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.
APPROXIMATIONS; LINEAR PROGRAMMING;NON- LINEAR FUNCTIONS; PROJECT MANAGEMENT WITH PERT/CPM; DECISION THEORY; THEORY OF GAMES; INVENTORY MODELLING; QUEUING THEORY
I am Irene M. I am a Diffusion Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from California, USA.
I have been helping students with their homework for the past 8 years. I solve assignments related to Diffusion. Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Diffusion Assignments.
APPROXIMATIONS; LINEAR PROGRAMMING;NON- LINEAR FUNCTIONS; PROJECT MANAGEMENT WITH PERT/CPM; DECISION THEORY; THEORY OF GAMES; INVENTORY MODELLING; QUEUING THEORY
I am Irene M. I am a Diffusion Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from California, USA.
I have been helping students with their homework for the past 8 years. I solve assignments related to Diffusion. Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Diffusion Assignments.
These are my note in the class of probabilistic analysis for the "average case" input. The look at:
1.- The use of the indicator function
2.- Enforcing the "Uniform Assumption"
At the end, we look at the application of the insertion sort average case.
Here is the first set of notes for the first class in Analysis of Algorithm. I added a dedicatory for my dear Fabi... she has showed me what real idealism is....
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
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journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
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A LEAST ABSOLUTE APPROACH TO MULTIPLE FUZZY REGRESSION USING Tw- NORM BASED O...ijfls
A least absolute approach to multiple fuzzy regression using Tw-norm based arithmetic operations is
discussed by using the generalized Hausdorff metric and it is investigated for the crisp input- fuzzy output
data. A comparative study based on two data sets are presented using the proposed method using shape
preserving operations with other existing method.
I am Lawrence B. I am a Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Masters's in Matlab from, Durham University, UK. I have been helping students with their assignments for the past 5 years. I solve assignments related to Signal Processing.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com. You can also call on +1 678 648 4277 for any assistance with Signal Processing Assignments.
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.
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.
I am Stacy L. I am a Matlab Assignment Expert at matlabassignmentexperts.com. I hold a Master's in Matlab, University of Houston. I have been helping students with their homework for the past 9 years. I solve assignments related to Data Analysis.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Data Analysis Assignments.
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 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.
These are my note in the class of probabilistic analysis for the "average case" input. The look at:
1.- The use of the indicator function
2.- Enforcing the "Uniform Assumption"
At the end, we look at the application of the insertion sort average case.
Here is the first set of notes for the first class in Analysis of Algorithm. I added a dedicatory for my dear Fabi... she has showed me what real idealism is....
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
A LEAST ABSOLUTE APPROACH TO MULTIPLE FUZZY REGRESSION USING Tw- NORM BASED O...ijfls
A least absolute approach to multiple fuzzy regression using Tw-norm based arithmetic operations is
discussed by using the generalized Hausdorff metric and it is investigated for the crisp input- fuzzy output
data. A comparative study based on two data sets are presented using the proposed method using shape
preserving operations with other existing method.
I am Lawrence B. I am a Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Masters's in Matlab from, Durham University, UK. I have been helping students with their assignments for the past 5 years. I solve assignments related to Signal Processing.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com. You can also call on +1 678 648 4277 for any assistance with Signal Processing Assignments.
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.
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.
I am Stacy L. I am a Matlab Assignment Expert at matlabassignmentexperts.com. I hold a Master's in Matlab, University of Houston. I have been helping students with their homework for the past 9 years. I solve assignments related to Data Analysis.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Data Analysis Assignments.
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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.
This lecture contains Newton Raphson Method working rule, Graphical representation, Example, Pros and cons of this method and a Matlab Code.
Explanation is available here: https://www.youtube.com/watch?v=NmwwcfyvHVg&lc=UgwqFcZZrXScgYBZPcV4AaABAg
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.
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.
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
Chi-squared Goodness of Fit Test Project Overview and.docxmccormicknadine86
Chi-squared Goodness of Fit Test Project
Overview and Rationale
This assignment is designed to provide you with hands-on experience in generating
random values and performing statistical analysis on those values.
Course Outcomes
This assignment is directly linked to the following key learning outcomes from the course
syllabus:
• Use descriptive, Heuristic and prescriptive analysis to drive business strategies and
actions
Assignment Summary
Follow the instructions in this project document to generate a number of different random
values using random number generation algorithm in Excel, the Inverse Transform. Then
apply the Chi-squared Goodness of Fit test to verify whether their generated values belong
to a particular probability distribution. Finally, complete a report summarizing the results
in your Excel workbook. Submit both the report and the Excel workbook.
The Excel workbook contains all statistical work. The report should explain the
experiments and their respective conclusions, and additional information as indicated in
each problem. Be sure to include all your findings along with important statistical issues.
Format & Guidelines
The report should follow the following format:
(i) Introduction
(ii) Analysis
(iii) Conclusion
And be 1000 - 1200 words in length and presented in the APA format
Project Instructions:
The project consists of 4 problems and a summary set of questions. For each problem, tom
hints and theoretical background is provided.
Complete each section in a separate worksheet of the same workbook (Excel file). Name
your Excel workbook as follows:
ALY6050-Module 1 Project – Your Last Name – First Initial.xlsx
In the following set of problems, r is the standard uniform random value (a continuous
random value between 0 and 1).
Problem 1
Generate 1000 random values r. For each r generated, calculate the random value 𝑿𝑿 by:
𝒙𝒙 = −𝑳𝑳𝑳𝑳(𝒓𝒓),
where “Ln“ is the natural logarithm function.
Investigate the probability distribution of X by doing the following:
1. Create a relative frequency histogram of X.
2. Select a probability distribution that, in your judgement, is the best fit for X.
3. Support your assertion above by creating a probability plot for X.
4. Support your assertion above by performing a Chi-squared test of best fit with a 0.05
level of significance.
5. In the word document, describe your methodologies and conclusions.
6. In the word document, explain what you have learned from this experiment.
Hints and Theoretical Background
A popular method for generating random values according to a certain probability
distribution is to use the inverse transform method. In this method, the cumulative
function of the distribution (F(x)) is used for such a random number generation. More
specifically, a standard uniform random value r is generated first. Most software
environments are capable of generating such a value. In E
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 Newton-Raphson method ( also known as Newton's method) is a way to quickly find a good approximation for the root of a real-valued function f (x) = 0f (x) = 0. It uses the idea that a nonstop and differentiable function can be approached by a straight line tangent to it.
This slide includes the basic descriptions and function determinations ideas.
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The International Journal of Engineering and Science (The IJES)
1. The International Journal Of Engineering And Science (IJES)
||Volume||2 ||Issue|| 11||Pages|| 05-12||2013||
ISSN(e): 2319 – 1813 ISSN(p): 2319 – 1805
On the Rate of Convergence of Newton-Raphson Method
Ranbir Soram1, Sudipta Roy2, Soram Rakesh Singh3, Memeta Khomdram4,
Swavana Yaikhom4 , Sonamani Takhellambam1
1
Manipur Institute of Technology, Takyelpat, Imphal-795004
Department of Information Technology, Assam University, Silchar-788011
3
Rajiv Gandhi Institute of Technology, Bangalore-560032
4
National Institute of Electronics and Information Technology, Akampat, Imphal-795001
2
------------------------------------------------------------ABSTRACT--------------------------------------------------------A computer program in Java has been coded to evaluate the cube roots of numbers from 1 to 25 using NewtonRaphson method in the interval [1, 3]. The relative rate of convergence has been found out in each calculation.
The lowest rate of convergence has been observed in the evaluation of cube root of 16 and highest in the
evaluation of cube root of 3. The average rate of convergence of Newton-Raphson method has been found
to be 0.217920.
KEYWORDS : Computer Program, Cube Root, Java, Newton-Raphson Method, Rate of Convergence.
-------------------------------------------------------------------------------------------------------------------------------------Date of Submission: 21, October, 2013
Date of Acceptance: 10, November, 2013
--------------------------------------------------------------------------------------------------------------------------------------
I. INTRODUCTION
Numerical analysis is a very important branch of computer science that deals with the study of
algorithms that use numerical approximation in mathematical analysis. It involves the study of methods of
computing numerical data. It has its applications in all fields of engineering and the physical sciences including
the life sciences. The stochastic differential equations and Markov chains have been used in simulating living
cells for medicine and biology. Numerical analysis computes numerical data for solving numerically the
problems of continuous mathematics. So, it implies producing a sequence of approximate values; thus the
questions involve the rate of convergence, the accuracy of the answer, and even time consumed to report the
answer. Many problems in mathematics can be reduced in polynomial time to linear algebra, this too can be
studied numerically. Root-finding algorithms are also used to solve nonlinear equations. If the function is
differentiable and the derivative is known and not equal to zero, then Newton-Raphson method is a popular
choice. Linearization is another technique for solving nonlinear equations. The formal academic area of
numerical analysis varies from quite foundational mathematical studies to the computer science issues involved
in the creation and implementation of several algorithms.
NEWTON-RAPHSON METHOD
In numerical analysis, Newton-Raphson method is a very popular numerical method used for finding
successively better approximations to the zeroes of a real-valued function x : f ( x) 0. Even though the method
can also be extended to complex functions, we shall restrict ourselves to real-valued functions only. NewtonRaphson method has been used by a large class of users as it works very well for a large variety of equations like
polynomial, rational, transcendental, trigonometric, and so on. It is also well suited on computers as it is
iterative in nature. This feature of Newton-Raphson method has attracted many scientists and many scientific
application programs use Newton-Raphson method as one of the root finding tools.The Newton-Raphson
method in one variable is implemented as follows: Given a function f ( x) defined over the real x, and its
derivative f '( x) , we begin with a first guess x0 for a root of the function f. Provided the function satisfies all the
assumptions made in the derivation of the formula, a better approximation x1 is
f ( x0 )
x1 x0
(1)
f '( x0 )
Geometrically, (x1, 0) is the intersection with the x-axis of a line tangent to f at (x0, f (x0)).
The process is repeated as
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2. On the Rate of Convergence of…
xn 1 xn
f ( xn )
(2)
f '( xn )
until a sufficiently desired accurate value is obtained.
The idea of the method is as follows: one starts with an initial guess which is reasonably close to the true root,
then the function is approximated by its tangent line (which can be computed using the tools of calculus), and
one computes the x-intercept of this tangent line (which is easily done with elementary algebra). This x-intercept
will typically be a better approximation to the function's root than the original guess, and the method can be
iterated. Suppose f : [a, b] R is a differentiable function defined on the interval [a, b] with values in the real
numbers R. The formula for converging on the root can be easily derived. Suppose we have some current
approximation xn. Then we can derive the formula for a better approximation, xn+1 by referring to the fig. 1 given
below. We know from the definition of the derivative at a given point that it is the slope of a tangent at that
point.
Fig. 1: Geometrical Interpretation of Newton-Raphson method
That is
f '( xn )
y
x
f ( xn ) 0
(3)
xn xn 1
Here, f ' denotes the derivative of the function f. Then by simple algebra we can derive
xn 1 xn
f ( xn )
(4)
f '( xn )
We start the iteration with some arbitrary initial value x0. The method will usually converge, provided
this initial guess is close enough to the unknown root, and that f '( x) 0. We reproduce here the algorithm for
Newton-Raphson method and it is from [2].
Algorithm 1. Newton-Raphson Method
1 Read x0, epsilon, delta, n
Remarks: Here x0 is the initial guess, epsilon is the prescribed relative error, delta is the prescribed lower bound
for f ' and n is the maximum number of iterations to be allowed. Statements 3 to 8 are repeated until the
procedure converges to a root or iterations equal n.
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3. On the Rate of Convergence of…
2
for i 1 to n in steps of 1 do
3
f 0 f ( x0 )
4
f 0 f 0 ( x0 )
5
if f 0 delta then goto 11
6
x1 x0
7
if
'
'
'
f0
'
f0
x1 x0
epsilon then goto 13
x1
8
x0 x1
endfor
'
9 Write “ does not converse in n iterations”, f 0 , f 0 , x0 , x1
10 Stop
'
11 Write “ slope too small ”, x0 , f0 , f0 , i
12 Stop
13 Write “ convergent solution”, x1 , f ( x1 ), i
14 stop
II. ESTIMATING THE RATE OF CONVERGENCE
Let us take a sequence x1 , x2 ,..., xn . If the sequence converges to a value r and if there exist real
numbers λ > 0 and α ≥ 1 such that lim
n
xn 1 r
xn r
(5)
then we say that α is the rate of convergence of the sequence. Many root-finding algorithms are in fact fixedpoint iterations. These iterations have the name fixed-point because the desired root r is a fixed point of a
function g(x), i.e., g (r ) r. For fixed-point iterations, if a value close to r is substituted into g(x) then the
result is even closer to r so we are tempted to write xn 1 g ( xn ).
It might be very convenient to define the error after n steps of an iterative root-finding algorithm as
en xn r . As n → ∞ we see from equation (5) that
en 1 en
en en 1
(6)
(7)
Dividing equation (6) by equation (7) we get the following formula given in equation (8).
en 1
en
en
en1
e
n
en 1
(8)
Now we solve for α and it gives
log
en 1
en
log
(9)
en
en 1
Using the formula given in equation (9), we can determine the convergence rate α from two consecutive
error ratios. But there is still a difficulty with this approach as to compute en we would need to know the root r
which in fact we don’t know. Anyway we can use the ideas above to develop a formula to estimate the rate of
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4. On the Rate of Convergence of…
convergence of any method that behaves like a fixed-point method. If the method converges, then the sequence
xn it generates must satisfy the relation xn1 r xn r , for n > N where N is some positive integer. As
mentioned above, to compute en xn r we need to know the value of the root r which we still do not know.
We will overcome this by assuming that the ratio of consecutive errors can be approximated by the ratio of
consecutive differences of the estimates of the root:
en1 xn1 r xn1 xn
en
xn r
xn xn 1
(10)
This allows us to approximate α with
log
xn 1 xn
xn xn 1
(11)
x xn 1
log n
xn 1 xn 2
Even though this only gives an estimate of α, in practice it agrees well with the theoretical convergence rates of
Newton’s method and gives us a good measure of the efficiency of various forms of fixed point algorithms.
Another formula that we can approximate the convergence rate α of Newton-Raphson method is
1
scaling factor X
(12)
No. of Iterations X f ( r )
where r is the estimated root and in case f (r ) 0, the formula fails.
III. WHY DO WE USE NEWTON-RAPHSON METHOD
The beauty of Newton-Raphson method lies in its efficiency and accuracy. The number of correct
digits in each intermediate solution roughly at least doubles in every iteration and thus, a very high accuracy is
obtained very quickly and it can be judged from the fig. 2. Thus, the use of Newton-Raphson method is highly
demanding. In a problem where the Newton-Raphson method converges in one iteration, it amounts to
f ( x0 )
using and evaluating one single formula: xroot x0
in order to obtain the desired root. Rather
f '( x0 )
than solving any equations the users are just evaluating functions. In practice a single evaluation of the
formula may fail to give the root, but two successive evaluations may give the solution. This again amounts
to using the formula twice and is still reasonable and attractive. Although the functions involved may not
converge so easily or the values of their derivatives at some points are too small to be used in the formula, a
sufficiently small neighborhood can be found out in which they are almost convergent and the values of their
derivatives at that neighborhoods can fit in the formula. By focusing our attention to such a neighborhood,
Newton-Raphson method would yield the desired root in two iterations, if not in one. The starting point plays a
very important role in the working of Newton-Raphson method. In general, a starting value far away from the
actual solution may take more iteration to get the desired root. Sometime it may also happen that the iterations
oscillate around the solution refusing to converge to it. There are many ways to avoid these conditions.
Generally, choosing the initial guess sufficiently close to the solution is attractive. The closer to the zero of the
function, the better it is. But, in the absence of any information about where the root might lie, a guess method
might narrow the possibilities to a reasonably small interval by using the intermediate value theorem.
Table 1: Rate of convergence of Newton-Raphson method in the calculation of roots of f(x)=x3-n=0.
S.No
Function
No. of
Iteration
1
f(x)=x3-1=0
1
2
f(x)=x3-2=0
7
3
f(x)=x3-3=0
8
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Value of the Root
1.000000000000000000000000000
00000000000000000000000000000
0000000000000
1.259921049894873164767210607
27822835057025146470150798008
1975112155300
1.442249570307408382321638310
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Rate of
convergence
Time taken
in ms
5.000000
0
1.058113
14
2.089114
16
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6. On the Rate of Convergence of…
23
f(x)=x3-23=0
11
24
f(x)=x3-24=0
11
25
f(x)=x3-25=0
11
9783733783750
2.843866979851565477695439400
95865185276416517273704810465
3425230235441
2.884499140614816764643276621
56021917678373850699870115509
2832389083375
2.924017738212866065506787360
13792277853049863510103004142
2573510072560
Average rate of convergence
0.072259
32
0.030560
31
0.155596
31
0.217920
IV. RESULT AND DISCUSSION
We have calculated the cube roots of integers from 1 to 25 using Newton-Raphson method in the
8
interval [1, 3] with an initial guess of 1 in each case. We have used equation (12) with a scaling factor of 1x10 .
“0.000000000000000000000000000000000000000000000000000000000000000000000001” is the tolerance
limit used in our program. The rate of convergence, value of roots of the functions, and time takens in
millisecond to obtain the roots are shown in Table-2. In the calculation of cube root of 1 and 8, we have taken α
as 5.0 as f ( r ) equals 0 in each case. As this will affect the average rate of convergence, we have excluded
them in the calculation of the average rate of convergence.
As an example, we choose to calculate a root of the equation f ( x) x 21 0. The graph showing
the value of each intermediate value at each iteration is given in fig. 2. The graph showing the time taken in
3
millisecond to calculate a zero of f ( x) x n 0 for n 1 to 25 is shown in fig. 3.The speed at which the
program reports a root of the function is highly attractive. In most cases, the program has taken less than 33
milliseconds to calculate a root of the function. It is highly surprising as we use
“0.000000000000000000000000000000000000000000000000000000000000000000000001” as the tolerance
limit in the program. We can improve the program a lot if we choose different initial guess for each function.
The program executed very quickly in the calculation of cube root of 2 and it had taken 14 milliseconds to report
a root. The most time consuming execution of the program was reported in the calculation of a cube root of 12 as
it took 1389 milliseconds to report the root. The graph in fig. 4 shows the relative rate of convergence in the
calculation of cube roots of integers from 1 to 25 of Newton-Raphson method in the interval [1, 3].
3
Fig. 2: Value of function f(x)=x3-21 from iteration 1 to 11
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7. On the Rate of Convergence of…
Fig. 3: Time taken in millisecond to calculate a root of f(x)=x3-n=0
V. CONCLUSION
The highest rate of convergence of Newton-Raphson method has been observed in the calculation of
cube root of 3 and is equal to 2.089114. The lowest rate of convergence of Newton-Raphson method has been
observed in the calculation of cube root of 16 and is equal to 0.000185. Average rate of convergence of NewtonRaphson method is calculated to be 0.217920.
Fig. 4: Rate of convergence in reporting a root of f(x)=x3-n=0
ACKNOWLEDGEMENTS
The first author thanks his daughter Java and son Chandrayan for not using his laptop during the
preparation of the manuscript. The fourth author thanks her husband for his understanding and support in the
preparation of the manuscript. The fifth author thanks her family and teachers for their supports in the writing of
this manuscript.
REFERENCES
[1]
[2]
[3]
[4]
[5]
Zoltan Kovacs, Understanding convergence and stability of the Newton-Raphson method (Department of Analysis, University of
Szeged, 2011).
V. Rajaraman, Computer Oriented Numerical Method (Prentice-Hall of India, 2005).
E Balagurusamy, Computer Oriented Statistical and Numerical Methods(Macmillan India Limited, 1988)
C. Woodford , C. Phillips, Numerical Methods with Worked Examples: Matlab Edition (Springer London , 1997).
N. Daili, Numerical Approach to Fixed Point Theorems, Int. J. Contemp. Math. Sciences, Vol. 3, 2008, no. 14, 675 – 682.
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8. On the Rate of Convergence of…
Ranbir Soram works at Manipur Institute of Technology,Takyelpat,
Imphal, India. His field of interest includes Network Security, NLP, Neural
Network, Genetic Algorithm, and Fuzzy Logic etc.
Sudipta Roy works in the Department of Information Technology, Assam
University Silchar.
Soram Rakesh Singh is a PG student in the Department of Computer
Science & Engineering at Rajiv Gandhi Institute of Technology, Bangalore.
Memeta Khomdram works at National Institute of Electronics and
Information Technology (Formerly DOEACC Centre), Akampat, Imphal.
Swavana Yaikhom works at National Institute of Electronics and
Information Technology, Akampat, Imphal.
Sonamani
Takhellambam
works
Technology,Takyelpat, Imphal, India.
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at
Manipur
Institute
of
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