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
In numerical analysis, Newton's method (also known as the Newton–Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function.
The method starts with a function f defined over the real numbers x, the function's derivative f', and an initial guess x0 for a root of the function f.
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
In numerical analysis, Newton's method (also known as the Newton–Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function.
The method starts with a function f defined over the real numbers x, the function's derivative f', and an initial guess x0 for a root of the function f.
Regula Falsi or False Position Method is one of the iterative (bracketing) Method for solving root(s) of nonlinear equation under Numerical Methods or Analysis.
BIRCH (balanced iterative reducing and clustering using hierarchies) is an unsupervised data-mining algorithm used to perform hierarchical clustering over, particularly large data sets.
Linear Algebra may be defined as the form of algebra in which there is a study of different kinds of solutions which are related to linear equations. In order to explain the Linear Algebra, it is important to explain that the title consists of two different terms. The very first term which is important to be considered in the same, is Linear. Linear may be defined as something which is straight. Linear equations can be used for the calculation of the equation in a xy plane where the straight lines has been defined. In addition to this, linear equations can be used to define something which is straight in a three dimensional perspective. Another view of linear equations may be defined as flatness which recognizes the set of points which can be used for giving the description related to the equations which are in a very simple forms. These are the equations which involves the addition and multiplication.
Computer Oriented Numerical Analysis
What is interpolation?
Many times, data is given only at discrete points such as .
So, how then does one find the value of y at any other value of x ?
Well, a continuous function f(x) may be used to represent the data values with f(x) passing through the points (Figure 1). Then one can find the value of y at any other value of x .
This is called interpolation
Newton’s Divided Difference Formula:
To illustrate this method, linear and quadratic interpolation is presented first.
Then, the general form of Newton’s divided difference polynomial method is presented.
These slides contain information about Euler method,Improved Euler and Runge-kutta's method.How these methods are helpful and applied to our questions are detailed discussed in the slides.
Regula Falsi or False Position Method is one of the iterative (bracketing) Method for solving root(s) of nonlinear equation under Numerical Methods or Analysis.
BIRCH (balanced iterative reducing and clustering using hierarchies) is an unsupervised data-mining algorithm used to perform hierarchical clustering over, particularly large data sets.
Linear Algebra may be defined as the form of algebra in which there is a study of different kinds of solutions which are related to linear equations. In order to explain the Linear Algebra, it is important to explain that the title consists of two different terms. The very first term which is important to be considered in the same, is Linear. Linear may be defined as something which is straight. Linear equations can be used for the calculation of the equation in a xy plane where the straight lines has been defined. In addition to this, linear equations can be used to define something which is straight in a three dimensional perspective. Another view of linear equations may be defined as flatness which recognizes the set of points which can be used for giving the description related to the equations which are in a very simple forms. These are the equations which involves the addition and multiplication.
Computer Oriented Numerical Analysis
What is interpolation?
Many times, data is given only at discrete points such as .
So, how then does one find the value of y at any other value of x ?
Well, a continuous function f(x) may be used to represent the data values with f(x) passing through the points (Figure 1). Then one can find the value of y at any other value of x .
This is called interpolation
Newton’s Divided Difference Formula:
To illustrate this method, linear and quadratic interpolation is presented first.
Then, the general form of Newton’s divided difference polynomial method is presented.
These slides contain information about Euler method,Improved Euler and Runge-kutta's method.How these methods are helpful and applied to our questions are detailed discussed in the slides.
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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.
APPROXIMATIONS; LINEAR PROGRAMMING;NON- LINEAR FUNCTIONS; PROJECT MANAGEMENT WITH PERT/CPM; DECISION THEORY; THEORY OF GAMES; INVENTORY MODELLING; QUEUING THEORY
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.
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Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
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June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...
The newton raphson method
1. TARUN GEHLOT (B.E, CIVIL, HONOURS)
The Newton-Raphson Method
Already the Babylonians knew how to approximate square roots. Let's consider the
example of how they found approximations to .
Let's start with a close approximation, say x1=3/2=1.5. If we square x1=3/2, we obtain
9/4, which is bigger than 2. Consequently . If we now consider 2/x1=4/3, its
square 16/9 is of course smaller than 2, so .
We will do better if we take their average:
If we square x2=17/12, we obtain 289/144, which is bigger than 2.
Consequently . If we now consider 2/x2=24/17, its square 576/289 is of
course smaller than 2, so .
Let's take their average again:
x3 is a pretty good rational approximation to the square root of 2:
but if this is not good enough, we can just repeat the procedure again and again.Newton
and Raphson used ideas of the Calculus to generalize this ancient method to find the
zeros of an arbitrary equation
Their underlying idea is the approximation of the graph of the function f(x) by the tangent
lines, which we discussed in detail in the previous pages.Let r be a root (also called a
"zero") of f(x), that is f(r) =0. Assume that . Let x1 be a number close
to r (which may be obtained by looking at the graph of f(x)). The tangent line to the graph
of f(x) at(x1,f(x1)) has x2 as its x-intercept.
2. TARUN GEHLOT (B.E, CIVIL, HONOURS)
From the above picture, we see that x2 is getting closer to r. Easy calculations give
Since we assumed , we will not have problems with the denominator being
equal to 0. We continue this process and find x3 through the equation
This process will generate a sequence of numbers which approximates r.This
technique of successive approximations of real zeros is called Newton's method, or
the Newton-Raphson Method.
Example. Let us find an approximation to to ten decimal places.
Note that is an irrational number. Therefore the sequence of decimals which
defines will not stop. Clearly is the only zero of f(x) = x2
- 5 on the interval
[1,3]. See the Picture.
3. TARUN GEHLOT (B.E, CIVIL, HONOURS)
Let be the successive approximations obtained through Newton's method. We
have
Let us start this process by taking x1 = 2.
It is quite remarkable that the results stabilize for more than ten decimal places after only
5 iterations!
Example. Let us approximate the only solution to the equation
In fact, looking at the graphs we can see that this equation has one solution.
4. TARUN GEHLOT (B.E, CIVIL, HONOURS)
This solution is also the only zero of the function . So now we see
how Newton's method may be used to approximate r. Since r is between 0 and , we
set x1 = 1. The rest of the sequence is generated through the formula
We have