This document discusses numerical methods for finding the roots of nonlinear equations. It covers the bisection method, Newton-Raphson method, and their applications. The bisection method uses binary search to bracket the root within intervals that are repeatedly bisected until a solution is found. The Newton-Raphson method approximates the function as a linear equation to rapidly converge on roots. Examples and real-world applications are provided for both methods.
This presentation will be very helpful to learn about system of linear equations, and solving the system.It includes common terms related with the lesson and using of Cramer's rule.
Please download the PPT first and then navigate through slide with mouse clicks.
How to handle Initial Value Problems using numerical techniques?
#WikiCourses
https://wikicourses.wikispaces.com/Topic+Initial+Value+Problems
https://eau-esa.wikispaces.com/Topic+Initial+Value+Problems
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
This presentation will be very helpful to learn about system of linear equations, and solving the system.It includes common terms related with the lesson and using of Cramer's rule.
Please download the PPT first and then navigate through slide with mouse clicks.
How to handle Initial Value Problems using numerical techniques?
#WikiCourses
https://wikicourses.wikispaces.com/Topic+Initial+Value+Problems
https://eau-esa.wikispaces.com/Topic+Initial+Value+Problems
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
How to find the roots of Nonlinear Equations?
Newton-Raphson method is not the only way!
How about a system of nonlinear equations?
#WikiCourses
https://wikicourses.wikispaces.com/Topic+Roots+of+Nonlinear+Equations
Numerical Method Analysis: Algebraic and Transcendental Equations (Non-Linear)Minhas Kamal
Numerical Method Analysis- Solution of Algebraic and Transcendental Equations (Non-Linear Equation). Algorithms- Bisection Method, False Position Method, Newton-Raphson Method, Secant Method, Successive Approximation Method.
Visit here for getting code implementation- https://github.com/MinhasKamal/AlgorithmImplementations/blob/master/numericalMethods/equationSolving/NonLinearEquationSolvingProcess.c
Created in 2nd year of Bachelor of Science in Software Engineering (BSSE) course at Institute of Information Technology, University of Dhaka (IIT, DU).
Presentation of third- and fifth-order optical nonlinearities measurement using the D4Sigma-Z-scan Method. I present a resolution of propagation equation in general case (with third- and fifth-order nonlinearities) and a numerical inversion.
This presentation is conclude with experimental results.
Numerical Methods was a core subject for Electrical & Electronics Engineering, Based On Anna University Syllabus. The Whole Subject was there in this document.
Share with it ur friends & Follow me for more updates.!
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.
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.
A NEW STUDY TO FIND OUT THE BEST COMPUTATIONAL METHOD FOR SOLVING THE NONLINE...mathsjournal
The main purpose of this research is to find out the best method through iterative methods for solving the nonlinear equation. In this study, the four iterative methods are examined and emphasized to solve the nonlinear equations. From this method explained, the rate of convergence is demonstrated among the 1st degree based iterative methods. After that, the graphical development is established here with the help of the four iterative methods and these results are tested with various functions. An example of the algebraic equation is taken to exhibit the comparison of the approximate error among the methods. Moreover, two examples of the algebraic and transcendental equation are applied to verify the best method, as well as the level of errors, are shown graphically.
A NEW STUDY TO FIND OUT THE BEST COMPUTATIONAL METHOD FOR SOLVING THE NONLINE...mathsjournal
The main purpose of this research is to find out the best method through iterative methods for solving the
nonlinear equation. In this study, the four iterative methods are examined and emphasized to solve the
nonlinear equations. From this method explained, the rate of convergence is demonstrated among the 1st
degree based iterative methods. After that, the graphical development is established here with the help of
the four iterative methods and these results are tested with various functions. An example of the algebraic
equation is taken to exhibit the comparison of the approximate error among the methods. Moreover, two
examples of the algebraic and transcendental equation are applied to verify the best method, as well as the
level of errors, are shown graphically.
A NEW STUDY TO FIND OUT THE BEST COMPUTATIONAL METHOD FOR SOLVING THE NONLINE...mathsjournal
The main purpose of this research is to find out the best method through iterative methods for solving the nonlinear equation. In this study, the four iterative methods are examined and emphasized to solve the nonlinear equations. From this method explained, the rate of convergence is demonstrated among the 1st degree based iterative methods. After that, the graphical development is established here with the help of the four iterative methods and these results are tested with various functions. An example of the algebraic equation is taken to exhibit the comparison of the approximate error among the methods. Moreover, two examples of the algebraic and transcendental equation are applied to verify the best method, as well as the level of errors, are shown graphically.
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.
Fractional pseudo-Newton method and its use in the solution of a nonlinear sy...mathsjournal
The following document presents a possible solution and a brief stability analysis for a nonlinear system,
which is obtained by studying the possibility of building a hybrid solar receiver; It is necessary to mention that
the solution of the aforementioned system is relatively difficult to obtain through iterative methods since the
system is apparently unstable. To find this possible solution is used a novel numerical method valid for one and
several variables, which using the fractional derivative, allows us to find solutions for some nonlinear systems in
the complex space using real initial conditions, this method is also valid for linear systems. The method described
above has an order of convergence (at least) linear, but it is easy to implement and it is not necessary to invert
some matrix for solving nonlinear systems and linear systems.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
3.
Equations that can be cast in the form of a
polynomial are referred to as algebraic equations.
Equations involving more complicated terms, such
as trigonometric, hyperbolic, exponential, or
logarithmic functions are referred to as
transcendental equations. The methods presented in
this section are numerical methods that can be
applied to the solution of such equations, to which
we will refer, in general, as non-linear equations. In
general, we will we searching for one, or more,
solutions to the equation, f(x) = 0.
3
Introduction
4.
Bisection method
Newton – Raph son method
Secant method
False position method, etc
4
Various numerical
methods find the roots
5.
The History of the Bisection Method
Although there is little concrete knowledge of the
development the bisection method, we can infer that
it was developed a short while after the Intermediate
Value Theorem was first proven by Bernard Bolzano
in 1817 (Edwards 1979). It appears that it was used
as a proof of an intermediate theorem to the general
proof Bolzano was developing for the Intermediate
Value Theorem
5
Bisection method
7.
If a function f(x) is continuous and there is a point a
that is negative and a point b that is positive then
there is a point c between (a,b) that equal zero. An
interval is always chosen in [a,b] which includes the
root somewhere within. That interval [a,b] is then cut
in half, and the half that contains the root is then
chosen. That new interval is then cut in half once
again, and the half that contains the root is chosen
once again. The bisection method repeats these steps
numerous times until the approximation is within a
certain degree
7
procedure
8. 8
Example 1
Consider finding the root of f(x) = x2 - 3. Let εstep = 0.01, εabs = 0.01 and start with the
interval [1, 2].
Table 1. Bisection method applied to f(x) = x2 - 3.
a b f(a) f(b)
c = (a +
b)/2
f(c) Update
new b −
a
1.0 2.0 -2.0 1.0 1.5 -0.75 a = c 0.5
1.5 2.0 -0.75 1.0 1.75 0.062 b = c 0.25
1.5 1.75 -0.75 0.0625 1.625 -0.359 a = c 0.125
1.625 1.75 -0.3594 0.0625 1.6875 -0.1523 a = c 0.0625
1.6875 1.75 -0.1523 0.0625 1.7188 -0.0457 a = c 0.0313
1.7188 1.75 -0.0457 0.0625 1.7344 0.0081 b = c 0.0156
1.71988
/td>
1.7344 -0.0457 0.0081 1.7266 -0.0189 a = c 0.0078
9. 9
2.
2. Find the root of x4-x-10 = 0
The graph of this equation is given in the figure.
Let a = 1.5 and b = 2
Iteration
No.
a b c f(a) * f(c)
1 1.5 2 1.75 15.264 (+ve)
2 1.75 2 1.875 -1.149 (-ve)
3 1.75 1.875 1.812 2.419 (+ve)
4 1.812 1.875 1.844 0.303 (-ve)
5 1.844 1.875 1.86 -0.027 (-ve)
So one of the roots of x4-x-10 = 0 is approximately 1.86
10.
One of the biggest advantages to the bisection
method is that it never diverges. Error also
decreases with each iteration. Therefore, as the
interval keeps splitting, the approximation gets closer
and closer to the desired root
10
advantages
11.
The biggest disadvantage of the bisection method is that
it converges slower than other methods and it cannot
depict multiple roots. Furthermore, if two roots lie close to
each other then the bisection method makes it difficult to
find both roots simultaneously. In the specific case of
f(x)=x2, the bisection method fails to converge on the root
(0,0). If a point a is chosen to the left of the zero and the
same point is taken to the right of the zero then the root
will not be found.
11
Disadvantages
12.
Shot Detection in Video Content for Digital Video Library -
The study presented the usage of bisection method for shot
detection in video content for the Digital Video Library (DVL).
DVL is a networked Internet application allowing for storage,
searching, cataloguing, browsing, retrieval, searching and uni-
casting video sequences. The browsing functionality can be
significantly facilitated by a fast shot detection process.
Experiments show that usage of the bisection method, allows
for accelerating shot detection about 3÷150 times (related to the
shot density). At the end of the paper two possible networked
applications are presented: a medical DVL developed for
elearning purposes and a hypothetical networked news
application
12
Real-Life Applications
13. 13
Locating and computing periodic orbits in molecular
systems - The Characteristic Bisection Method for finding
the roots of non-linear algebraic and/or transcendental
equations is applied to Li NC/Li CN molecular system to
locate periodic orbits and to construct the
continuation/bifurcation diagram of the bend mode
family. The algorithm is based on the Characteristic Poly
hidra which define a domain in phase space where the
topological degree is not zero. The results are compared
with previous calculations obtained by the Newton
Multiple Shooting algorithm. The Characteristic Bisection
Method not only reproduces the old results, but also,
locates new symmetric and asymmetric families of
periodic orbits of high multiplicity.
Bisection method for determining an adequate
population size
15.
The name "Newton's method" is derived from Isaac Newton's
description of a special case of the method in De analysi per
aequationes numero terminorum infinitas (written in 1669,
published in 1711 by William Jones) and in De metodis
fluxionum et serierum infinitarum (written in 1671, translated
and published as Method of Fluxions in 1736 by John Colson).
However, his method differs substantially from the modern
method given above: Newton applies the method only to
polynomials.
15
History
16. 16
• He does not compute the successive approximations x_n, but computes a
sequence of polynomials, and only at the end arrives at an approximation for
the root x. Finally, Newton views the method as purely algebraic and makes
no mention of the connection with calculus. Newton may have derived his
method from a similar but less precise method by Vieta. The essence of
Vieta's method can be found in the work of the Persian mathematician
Sharaf al-Din al-Tusi, while his successor Jamshīd al-Kāshī used a form of
Newton's method to solve x^P - N = 0 to find roots of N (Ypma 1995).
• Newton's method was first published in 1685 in A Treatise of Algebra both
Historical and Practical by John Wallis. In 1690, Joseph Raphson published a
simplified description in Analysis aequationum universalis. Raphson again
viewed Newton's method purely as an algebraic method and restricted its
use to polynomials, but he describes the method in terms of the successive
approximations xn instead of the more complicated sequence of
polynomials used by Newton. Finally, in 1740, Thomas Simpson described
Newton's method as an iterative method for solving general nonlinear
equations using calculus, essentially giving the description above. In the
same publication, Simpson also gives the generalization to systems of two
equations and notes that Newton's method can be used for solving
optimization problems by setting the gradient to zero.
17. 17
Unlike the earlier methods, this method requires only one
appropriate starting point as an initial assumption of the root of
the function At a tangent to is drawn.
Equation of this tangent is given by
• The point of intersection, say , of this tangent with x-axis (y = 0)
is taken to be the next approximation to the root of f(x) = 0. So on
substituting y = 0 in the tangent equation we get
18. )( 00 xfy and
10
0
0
xx
y
x
dx
dy
atWe have and we need to find .
1x
Then,
10
0
0
/ )(
)(
xx
xf
xf
Rearranging:
)(
)(
0
/
0
10
xf
xf
xx
)(
)(
0
/
0
01
xf
xf
xx
Using and in the formula isn’t very
convenient, so, since we have)(xfy
0at x
dx
dy
0y
)( 0
/
10
0
0 xf
xx
y
x
dx
dy
at
19. )(
)(
0
/
0
01
xf
xf
xx So,
We just need to alter the subscripts to find : 2x
)(
)(
1
/
1
12
xf
xf
xx
Generalising gives
)(
)(
/1
n
n
nn
xf
xf
xx
We don’t need a diagram to use this formula but we
must know how to differentiate . )(xf
Convergence is often very fast.
21. 21
We will use the Newton-Raphson method to find the positive root of the equation sinx = x2,
correct to 3D.
It will be convenient to use the method of false position to obtain an initial approximation.
Tabulating, one finds
With numbers displayed to 4D, we see that there is a root in the interval 0.75 < x < 1
at approximately
Example: 1
22. 22
Next, we will use the Newton-Raphson method; we have
and
yielding
Consequently, a better approximation is
Repeating this step, we obtain
so that
Since f(x2) = 0.0000, we conclude that the root is 0.877 to 3D.
23.
The method is very expensive - It needs the function
evaluation and then the derivative evaluation.
If the tangent is parallel or nearly parallel to the x-axis,
then the method does not converge.
Usually Newton method is expected to converge only
near the solution.
The advantage of the method is its order of convergence
is quadratic.
Convergence rate is one of the fastest when it does
converge.
23
Advantages and
Disadvantages
24.
Applying NR to the system of equations we find that at
iteration k+1:
all the coefficients of KCL, KVL and of BCE of the linear
elements remain unchanged with respect to iteration k
Nonlinear elements are represented by a linearization of
BCE around iteration k
This system of equations can be interpreted as the STA of
a linear circuit (companion network) whose elements are
specified by the linearized BCE.
APPLICATION OF NEWTON RAPHSON METHOD TO
A FINITE BARRIER QUANTUM WELL (FBQW)
SYSTEM
Real life uses