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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.

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Non-periodic orbit in Newton-Raphson example

Let $f(z)=\frac{z^2-1}{2z}$. I want to find a point $\xi\in\mathbb{R}$ such that the $f$-orbit of $\xi$ is non-periodic and$$ \lim_{n\to\infty} \frac1n \#\{t\leq n \,:\,f^t(\xi)<0 \}=\frac13. $$ ...
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Creating variable functions using MATLAB

So I have three seperate function in MATLAB where each have its designated purpose. The first one calculates the partial derivative The second finds the roots for a system of two equations and two ...
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Find a critical point for $f(x,y)$ using Newton Raphson [MATLAB]

I was tasked with finding the critical point of a function $f(x,y)=(x-1)^2 + (y-1)^2$ Using my Newton Raphson program written in MATLAB. ...
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Finding a stationary point using Newtons Method for two variables.

So I've written a function in MATLAB which solves a non-linear system of equation: $f(x,y) = 0$ $g(x,y) = 0$ for two variables. I am now tasked with finding a stationary point (critical point) ...
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1answer
33 views

How do I iterate/increment inorder to find the roots for a system of equations with two variables using the Newton Method

As stated in the title I am interested in finding my roots for a system of two equations and two variables i.e: $f(x,y)=0$ $g(x,y)=0$ for a the functions: $f(x,y) =x(1+x^2)-1 $ & $g(x,y) = y(...
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1answer
34 views

Newtons Method with an unknown method

I am attempting to use Newtons method in an optimisation problem. I haven't used Newtons method since my college days and after refreshing my memory I am encountering the following challenge. ...
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1answer
21 views

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}{\...
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24 views

Convergence of Newton's method (prove)

How can I prove that: $$\Phi'(x) = 1- \frac{{[f '(x)]^2} - f(x).f ''(x)}{[f '(x)]^2} \Rightarrow \vert \Phi '(x) \vert \lt 1$$ Knowing that $$\Phi(x) = x -\frac{f(x)}{f '(x)} $$ by the Newton's method ...
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2answers
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If $N(x) = x-[f^{\prime}(x)]^{-1}f(x)$ and $f(x^\ast)=0$, how can I prove that $N^\prime(x^\ast) = 0$?

Let $\Omega\subseteq \mathbb{R}^n$ open in $\mathbb{R}^n$ and $f\in C^{2}(\Omega,\mathbb{R}^n)$. Let $N(x) = x-[f^{\prime}(x)]^{-1}f(x)$, supose that: there exist $x^\ast\in \Omega$ such that $f(x^\...
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Information about Newton's method at a certain step

We are preforming Newton's method with step $F'(x_n)\Delta x=-F(x_n)$, $x_{n+1}=\Delta x +x_n$, where $F:\mathbb{R}^n \rightarrow \mathbb{R}^n$. We can assume that it will converge. At a give time we ...
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1answer
41 views

Fast method to solve transcendental equation for a range of parameters?

I have an equation of the form $t + e^{Ax} + e^{Bx} = 0$ This is a transcendental equation, and I would use a Newton-Raphson algorithm or uniroot in ...
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1answer
37 views

On the convergence rate of Newton's method

Is that possible for some function whose convergence rate is linear by using Newton's method? I am solving the function $$f(x) = \sin^2(x) - x \sin(x) + \frac 14 x^2$$ by Newton's Method. I got the ...
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Show that two step Newton's method converges at least cubically?

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be $C^2$ for all x in a neighborhood of a non-degenerate root $x_*$, i.e., $f(x_*)=0$ with $f'(x_*) \neq 0$. Consider the sequence generated by $$ z = x_k - ...
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1answer
33 views

Minimizer of a multivariable function and iteration through Newton's method

I got stuck on the following question. Find the minimizer for $$f (x_1,x_2) = \frac 12 (x_1^2 - x_2)^2 + \frac 12 (1-x_1)^2$$ and compute one iteration for minimizing $f$ from point $(2,2)$. Also, ...
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3answers
59 views

Stopping criterion for Newton-Raphson

I am currently doing a question on Newton-Raphson method and I am not sure what it means by 'explain your stopping criterion'. Question Using the Newton-Raphson method with initial guess $x_0=1.5$,...
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1answer
34 views

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 ...
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34 views

How can I prove that (under certain conditions) the true root of a function lies between the Newton and Secant estimate?

I am new to Numerical methods, so please bear with me and try to explain it with this in mind.
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Are there extensions of Darboux theorem for higher order methods?

In $1869$, Darboux proved that, when using Newton method, there will never be an overshoot of the solution if the starting point $x_0$ is such that $$f(x_0) \times f''(x_0) >0$$ and, from a ...
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Do BFGS and L-BFGS methods converge when the matrix $H=B^{-1}$ is ill-conditioned?

I am using BFGS and L-BFGS to solve an unconstrained optimization problem. The objective function is the Mean Euclidean Error. The output is given by an Artificial Neural Network. The line search ...
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Pattern of Newton-Raphson iteration $x\mapsto\frac{1}{2}(x+\frac{q}{x})$ over finite fields

While playing with Newton-Raphson method over finite field $\mathbb{F}_p$, I noticed some cute patterns that I can't explain out of my brain contaminated with analysis. Here is the setting: ...
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2answers
60 views

Is modified Newton's Raphson method redundant?

I have been recently taught Newton's method for finding roots of non-linear equations. I was told in class that if the multiplicity of the root is more than 1, then the order of convergence is not ...
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Why does my newton-Raphson iteration fail?

Suppose that I have the following energy equation that is a function of $\varepsilon$, the strain, and $\eta$, the hardening parameter. $\phi=\frac{1}{2}E \varepsilon_e^2+\frac{1}{2}H \eta^2$, $\...
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1answer
36 views

Verify that the function $f(x) = e^{2x} - 2x - 1$ has a zero (root) of multiplicity 2 in 0.

From a previous problem, it is given that the function $f(x) = e^{2x} - 2x - 1$ has a zero of multiplicity two in zero. Using that information, I am trying to resolve the following problem: Using ...
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1answer
76 views

Obtain an aproximation to $\sqrt{5}$ using other numeric methods

From the original problem: Find an approximation to $\sqrt{5}$ correct to an exactitude of $10^{-10}$ using the bisection algorithm. In which I have a function in Mathematica to do the ...
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1answer
37 views

Newton's finding root method - n digits accuracy

I have read about the Newton's method in Wikipedia. https://en.wikipedia.org/wiki/Newton%27s_method By checking the pseudocode in the thread, I have noticed a tolerance is defined (in that case $10^...
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49 views

Complexity of the Newton's method approximation of $\frac{R}{b}$

I am trying to understand a particular concept in this lecture. The lecture goes on to describe that to compute a division of $\frac{a}{b}$, you have to: Compute high-precision representation of $\...
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37 views

Using Newton Raphson's method to solve a non-linear boundary value problem?

(My specific question is at the end of the problem) Examine the boundary problem with a nonlinear right hand side $1+u^2(x)$ $$-u''(x) = 1+u^2(x) \quad \text{on} \quad 0 < x < 1 \quad \text{...
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Using Newton-Raphson method, find the solution for $e^{\frac{x^2}{4vt}} = 1+\frac{x^2}{2vt}$

I need help with solving this difficult fluid dynamic expression. I have tried using rules of logs, symbolab algebra calculator and Wolfram Alpha calculator, and I have got no solution. How would ...
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What is the best method to use to find the root of a log shaped function?

I am trying to find the best method to get the root of a log shaped function. I have a function which is shaped very much like $y=ln(x)$. That is the part to the left of the zero is very steep and ...
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1answer
71 views

Matlab code for $y_{n+2} - y_n = h\left[(1/3)f_{n+2} + (4/3)f_{n+1} + (1/3)f_n\right]$

I am looking for a Matlab code for the $2$ stage multistep method : $$y_{n+2} - y_n = h\left[(1/3)f_{n+2} + (4/3)f_{n+1} + (1/3)f_n\right]$$ Note that $f_n = f(x_n,y_n)$. Now, since $f_{n+2} = f(x_{n+...
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dividing by variance to avoid trivial solutions in multidimensional Newton's method

I'm using Newton's method as implemented by GSL (GNU Scientific Library) to try to find a cycle of a multi-variate function $F$ (reaction-diffusion simulation over two-channel 2D images collapsed to ...
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1answer
58 views

Global convergence with Newton method for nonlinear systems

Let $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ a function for which we seek a fixed point, $$f:(x,y)\mapsto\begin{pmatrix} -5x+2\sin x+2\cos y\\ -5y+2\sin y+2\cos x \end{pmatrix}. $$ I made a python ...
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1answer
29 views

how to detect the points on which Newton Raphon method will give a oscillating sequence.

Example consider the function $x^3 -x/2 +1/4$.. we have a oscillation if we start from 0 or 0.5 in Newton Raphson Sequence. Why this oscillation pattern observed and how to detect it?
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Using Newton's method to find $M$, the unique positive zero of $x^2+x-1$

Using Newton's method to find $M$, the unique positive zero of $x^2+x-1$. Obtain a recursive formula for the error term $e_n$ use it to prove $a_n \rightarrow M$ Recursive formula for $a_n:\quad$ $...
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1answer
25 views

Newton's method, neighborhood of convergence

For Newton's method, my book says that for convergence, the starting point, $x^{(0)}$, must be sufficiently close to $x^*$, the actual root. According to the following inequality, where $C = \frac{...
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Newton iteration of non linear equation

I would like to find the roots of the problem $$z - y_i - (h/2 * (f(z))) = 0, $$ where z is a vector of length N and h is a constant (NB: $z = (y_{i+1} + y_{i})/2)$). $f(z) = -(z_{i+1} - z_{i-1})/...
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Do multivariate Householder methods exist?

Newton's method can be extended to higher-order versions using Householder's method. Newton's method can also be extended to the case of multivariate inputs, sometimes called the "Newton-Raphson ...
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Convergence of damped Newton’s method [closed]

Let $f$ be a twice continuously differentiable function satisfying $LI \succeq \nabla^2 f \succeq mI$ for some $L > m > 0$ and let $x^*$ be the unique minimizer of $f$ over $\Re^n$. Proof that ...
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Convergence of the Newton-Raphson method applied to a nonlinear system

I'm working on a nonlinear system of 2 equations and try to solve it with the Newton-Raphson method. I have a function $f(x,y)$. In order to use the method, I have to find a starting vector $(x,y)$. ...
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3answers
41 views

Reciprocal using Newton Raphson

Say you want to calculate 1/R using Newton-Rapshon method. Then we let, $$f(x) = 1/x - R$$ This means the root of the this function is at $f(1/R)$. So to find $1/R$, you can find the root of this ...
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235 views

Find good starting candidates for Newton-Raphson knowing one of the solutions of a parametrized system of nonlinear equations

I have a parameterized system of equations describing the crossed ladders problem. $(x, y)$ are the $2$ horizontal distances respectively on the left/right of the junction of the ladders $(a, b, c)$ ...
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Calculation of matrix according to Schubert's method

Schubert's method is the improvement of Broyden's method for the calculation of the quasi-newton update of a jacobian matrix, when the matrix itself is sparse. According to his paper, the update ...
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Prove Newton’s method works for finding roots of polynomials even if the polynomial’s derivative vanishes at the root.

Let $p(x)$ be a polynomial with real coefficients, and assume that $a = p(0) \neq 0$. Let $f(x) = x^mp(x)$ for some integer $m\geq2$. Let $x_0\in \mathbb{R}$ be a real number, and define $x_{n+1} = ...
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Quasi-Newton Methods no-change Condition Requirement

In standard quasi-newton methods for fixed point iteration, it looks there is two required conditions. The first one is secant condition: $$J_{k+1} \Delta x_{k} = \Delta f_k $$ where $\Delta f_k = f(...
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Why does calculating $\exp z$ using $\ln z$ via newton-raphson method fail to converge?

I am trying to calculate $\exp z$ using $\ln z$ via Newton-Raphson method $$x_{n+1} = x_n-\frac{f(x_n)}{f^{'}(x_n)}$$and got the formula $$x_{n+1}=x_n-\frac{\ln x_n-z}{\frac{1}{x_n}}$$ where $z = a + ...
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1answer
29 views

Newton-Raphson on strictly convex function

Maybe someone can give me a hint here: question 1: Given a sequence {$x_n$} which is the Newton-Raphson sequence on some $f(x)$ s.t. $f(x)$ is strictly convex and $f'>0$. Let $\alpha$ be the ...
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2answers
38 views

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 $...
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f is strongly convex and twice continuously differentiable. Is the Newton method guaranted to converge for any starting point and any step size?

This doesn't seem to be a theorem, but I can't find any counterexample. Let $f$ be a function in $\mathcal{R}^n \rightarrow \mathcal{R}$, strongly convex and twice continuously differentiable. Let'...
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1answer
52 views

How do I solve this problem without a given equation?

Do I need an equation to figure out $x_2$ and $x_3$ values? Can the equation be obtained from the graph? Much thanks!
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37 views

Defining a Jacobian Matrix

reaching out my post. I have a trouble of defining a Jacobian matrix for my problem. Basically, I have 4 differential equations to be solved. $\dot x_1(t)=x_2(t),$ $\dot x_2(t)=p_2(t)−\sqrt 2 x_1(t)...