Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [newton-raphson]

This tag is for questions regarding the Newton–Raphson method. In numerical analysis the Newton–Raphson method is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function.

The Newton–Raphson method in one variable is implemented as follows:

The method starts with a function $f$ defined over the real numbers, the function's derivative $f'$, and an initial guess $x_0$ for a root of the function $f$. If the function satisfies the assumptions made in the derivation of the formula and the initial guess is close, then a better approximation $x_1$ is $$x_1 = x_0 -\frac{f(x_0)}{f'(x_0)}$$

The process is repeated as

$$x_{n+1} = x_n -\frac{f(x_n)}{f'_n(x_n)}$$

until a sufficiently accurate value is reached.

798 questions
9
votes
1 answer

Reference request for Newton's method on a convex/concave function

The proof featured at http://planetmath.org/newtonsmethodworksforconvexrealfunctions shows that, roughly speaking, Newton's method is unconditionally convergent for convex/concave functions. I am looking for a textbook or paper reference of this…
parsiad
  • 25,154
3
votes
1 answer

Newton method with inequality constraints

im kinda confused and searching for the right method i need to use. I got a function $F:\mathbb{R}^3\rightarrow \mathbb{R}^3$ for which i need to solve $F(n_1,n_2,\theta)=0$ but with the constraint $||n||\le 1$. I first forgot about the constraint,…
korzi
  • 53
3
votes
1 answer

Newton Raphson Method in GPS

I'm writing a paper on GPS and how coordinates are found using triangulation methods. To find the coordinates on a 3D system, the Newton Raphson Method is needed. How would I do this and could an example be given as well? This is the equation for…
sousheel
  • 143
  • 1
  • 5
3
votes
2 answers

How does Newton's Method work?

Before I am told, I want to clarify that I searched first, and I don't believe this to be a repost. I understand the formula in terms of how to apply it, and I've seen graphical representations and everything. I get that we are finding where the…
Alex G
  • 237
2
votes
0 answers

Newton's method for a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$

For a differentiable function $f:\mathbb{R} \rightarrow \mathbb{R}$, Newton's method consists of iterating $$x_{n+1} = x_{n} - \frac{f(x_n)}{f'(x_n)}$$ where $x_0$ is some initial guess, to find a point $f(x^*) \approx 0$, and $x_i \in…
IntegrateThis
  • 3,778
2
votes
2 answers

Distance to a curve via Newton's Method

So I wanted a quick confirmation if my thinking is correct. The particular problem I'm working on is a $2$D Bezier Curve, with at most $4$ control points and at a minimum of $2$ control points. The more general question (also I'm going to use the…
Wesley Stover
  • 21
2
votes
1 answer

Why $\det[J(x_i , y_i, ..., z_i)]$ need to be different from $0$?

In matrix form of Newton's method for system of non-linear equations: $$\begin{bmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} & \cdots & \frac{\partial f_1}{\partial z} \\ \frac{\partial f_2}{\partial x} & \frac{\partial…
user655901
2
votes
3 answers

Constructing Newton iteration converging to non-root

Is it possible to construct a Newton sequence $x_{n+1} := x_{n} - f(x_n)/f'(x_{n})$ such that $\{x_{n}\}$ is a Cauchy sequence converging to $x^*$, but $x^{*}$ is not a root of $f$? (Perhaps because $f$ has no roots?)
user14717
  • 4,902
2
votes
1 answer

Convergence of ratios of successive terms in Newton's method

I was experimenting with ratios of success iterations of Newton's method for a class assignment, and I noticed that for $x_n$ near a root $x^*$ of $f$, for which $f(x^*)\neq 0$,…
user3002473
  • 8,943
2
votes
3 answers

Finding reciprocal via Newton Raphson: How to determine initial guess?

I'm learning Newton-Raphson to get the reciprocal of any arbitrary value. I need this in order to accurately emulate how the PlayStation 1 does the divide. They used a modified version of the algorithm. Trying to understand the basic algorithm I…
vexe
  • 189
1
vote
0 answers

How to show that Newton's method is convergent/divergent?

I have a function $g(x)=-e^{-x^2}$. The task is to use a Newton’s method on the function $g'$ and find a value $\alpha$ such that if $x_0 \in [0,\alpha)$ the Newton's method converges to $0$ and if $x_0>\alpha$ the Newton's method diverges. I was…
delta
  • 13
1
vote
0 answers

On a sufficient condition for the convergence of the Newton-Raphson method

Given a suitable function $f$, often of class $C^2$, in the section of quadratic convergence for Newton's iterative method mentioned in WikipediA I found the following estimate $$ \displaystyle \left|{\varepsilon _{n+1}}\right|={\frac {\left|f''(\xi…
QA Ngô
  • 466
1
vote
0 answers

Newton's method with the second derivative included

The usual formulation of Newton's method goes like - $f(x) = f(a) + (x-a)f'(a)$ As $x$ is a root of $f$, $f(x) = 0$ And so - $x = a - \frac{f(a)}{f'(a)}$ I was wondering if there was any way to include the second derivative too. $f(x) = f(a) +…
Paddy
  • 457
1
vote
1 answer

Linear Function and Newton's Method

I want to show that if a function is linear for example F(x) = 3x + 3, then after one iteration of Newton's method I can find the x-value such that F(x) = 0. In this case, after one iteration x = -1. I see that it works after one iteration, but how…
David Zheng
  • 61
1
vote
1 answer

Newton Raphson Application from paper

I just want to double check that I'm understanding the method followed in this paper. They provide a flow chart with the method followed here. The equations used are provided here: Specifically equation 12,18,19. I'm doing the procedure using…
Engjunkie
  • 11
1
2 3 4