newton raphson method algorithm

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The Newton-Raphson Method

The Newton-Raphson method or Newton Method is a powerful technique for solving equations numerically Like so much of the di erential calculus it is based on the simple idea of linear approximation The Newton Method properly used usually homes in on a root with devastating e ciency

How to solve a non-linear equation using Newton Raphson method?
An algorithm for Newton Raphson method requires following steps in order to solve any non-linear equation with the help of computational tools: 1. Start 2. Define function as f (x) 3. Define first derivative of f (x) as g (x) 4. Input initial guess (x0), tolerable error (e) and maximum iteration (N) 5. Initialize iteration counter i = 1 6.
What is an algorithm for Newton Raphson method?
And an algorithm for Newton Raphson method involves repetition of above process i.e. we use x1 to find x2 and so on until we find the root within desired accuracy. An algorithm for Newton Raphson method requires following steps in order to solve any non-linear equation with the help of computational tools:
Is Raphson a linear approximation?
For polynomials, Raphson’s procedure is equivalent to linear approximation. Raphson, like Newton, seems unaware of the connection between his method and the derivative. The connection was made about 50 years later (Simpson, Euler), and the Newton Method nally moved beyond polynomial equations.

PDF	STABILITY ALGORITHMS FOR NEWTON-RAPHSON METHOD IN ABSTRACT. This paper deals with possible algorithms which may ensure numerical stability of Newton-Raphson method in load flow analysis.

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PDF	Appendix A - The Newton-Raphson Algorithm Algorithm. The Newton-Raphson algorithm is a commonly used technique for locating zeros of a function. Let H:IRn --+ IRn have a zero at x* that is

PDF	Parallel Hybrid Algorithm of Bisection and Newton-Raphson It is organized as follows: section 2 establishment to root-finding Bisection

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PDF	An Improved Hybrid Algorithm to Bisection Method and Newton 09-Nov-2017 On the other hand the Newton-Raphson method using the derivative of a given nonlinear function is a root-finding algorithm which is more ...

PDF	Generalized Newton Raphsons method free from second derivative In this section we will show that the convergence order of modified generalized Newton Raphson's method (Algorithm 3.1) is at least six and that of generalized

PDF	Appendix A - Solving Systems of Nonlinear Equations A.1 Newton-Raphson Algorithm. The Newton-Raphson algorithm is described in this section. A.1 Algorithm flowchart for the Newton-Raphson method.

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PDF	Improvements in Newton-Rapshon Method for Nonlinear Equations 23-Jul-2015 Abstract. In this paper we present two new numerical algorithms for solv- ing nonlinear equations based on Newton-Raphson method. New al-.

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Newton-Raphson Method Algorithm

The Newton-Raphson method is an iterative numerical technique used for finding successively better approximations to the roots (or zeroes) of a real-valued function. It's widely used in various fields, including engineering, physics, and mathematics.

This tutorial provides a comprehensive overview of the Newton-Raphson method algorithm, including examples, exercises, case studies, and important notes to enhance understanding and application.

Examples

1. Finding the square root of a number using the Newton-Raphson method.

2. Solving a nonlinear equation such as \(x^2 - 4x + 3 = 0\) using the Newton-Raphson method.

3. Calculating the inverse square root using the Newton-Raphson method.

4. Estimating the solution to a transcendental equation like \(e^x - x - 2 = 0\).

5. Approximating the solution to a trigonometric equation such as \(\sin(x) - x = 0\).

Exercises with Solutions

Find the square root of 5 using the Newton-Raphson method.

Solution:

Let \(f(x) = x^2 - 5\). Applying the Newton-Raphson formula:
\(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\)

Starting with an initial guess \(x_0 = 2\),
\(x_1 = 2 - \frac{2^2 - 5}{2 \times 2} = \frac{7}{4} = 1.75\)
\(x_2 = 1.75 - \frac{1.75^2 - 5}{2 \times 1.75} \approx 2.0714\)

Iterating further, we converge to \(x \approx 2.2361\).

Correct Answer: \(x \approx 2.2361\)

Multiple Choice Questions (MCQ) with Answers

What is the key idea behind the Newton-Raphson method?
a) Linear interpolation
b) Iterative approximation
c) Matrix inversion
d) Polynomial regression
Correct Answer: b) Iterative approximation

Case Studies

Case Study 1: Application of the Newton-Raphson method in calculating the electrical power output of a solar panel based on varying sunlight intensity.

Case Study 2: Use of the Newton-Raphson method to determine the optimal shape of a parabolic mirror for focusing light in a solar concentrator system.

Case Study 3: Implementation of the Newton-Raphson method in optimizing the trajectory of a spacecraft for interplanetary missions.

Case Study 4: Application of the Newton-Raphson method in financial modeling for estimating future stock prices based on historical data.

Case Study 5: Use of the Newton-Raphson method in medical imaging for reconstructing three-dimensional structures from two-dimensional X-ray images.

Important Notes

The Newton-Raphson method requires an initial guess close to the root for convergence.
Convergence is not guaranteed for all functions or initial guesses, and oscillation or divergence may occur.
The method may require fewer iterations and provide faster convergence compared to other root-finding techniques.
The choice of initial guess and the derivative function significantly impact the efficiency and accuracy of the method.
The Newton-Raphson method is widely used in computer algorithms, scientific simulations, and engineering design.

PDF	Algorithme sur la méthode Newton-Raphson - Lycée dAdultes 5 nov 2015 · Algorithme sur la méthode Newton-Raphson 1 Historique La méthode de résolution des équations numériques a été initiée par Isaac New-

PDF	The Newton-Raphson Method - UBC Math For example, by putting a little bump on the curve at x = a we can make b fly far away from r When a Newton Method calculation is going badly, a picture can help

PDF	The Newton Raphson Algorithm for Function Optimization The Newton Raphson algorithm is an iterative procedure that can be used to approximation to the function of interest around some initial parameter value (

PDF	Generalized Newton Raphsons method free from second - EMIS Algorithm 3 2 is called the generalized Newton Raphson's method free from second derivative With the help of this method, we can solve such type of non linear

PDF	On Newton-Raphson formulation and algorithm for displacement On the basis of a full Newton-Raphson method, a specific source of nonlinearity arised from quadratic damping, also known as quadratic velocity or hydrodynamic

PDF	NEWTONS METHOD AND FRACTALS 1 - Whitman College example, to solve for the roots of a quadratic function ax2 + bx + c = 0 we may the Newton-Raphson method, or more commonly Newton's method [3]