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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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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
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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
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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
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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
33 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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38 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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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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5answers
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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
62 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
38 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
27 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
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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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0answers
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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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2answers
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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
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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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1answer
226 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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1answer
81 views

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
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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
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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
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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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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)...
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Rate of convergence of Newton method in changing variable

For example, I want to solve this equation for $x>0$ $$\sqrt{1-x^2} = 0.5 \tag{1}$$ I used Newton-Raphson method and it works nicely. I wonder if I set $x=\cos(\theta)$, (1) becomes $$\sin(\theta)...
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How can Newton's method ever work in neural netowrks?

In neural networks, we try to find successively better weights for the network by trying to minimize some error function. Gradient descent can be used. The error function may not be convex with ...
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Getting the inverse of a matrix with matrix elements

I am solving a problem regarding newton's method. We are using this as the function: click to see function Thus these are the Jacobian Matrix and the set up for Newton's Method: click here to see ...
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2answers
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What is the benefit of using forward difference approximation in newton's method of root finding?

I am trying to think of when using forward difference approximation to $$f'(x) = \frac{f(x+\delta) - f(x)}{\delta}$$ in Newton's root finding method of $$f(x_{n+1})=x_n-\frac{f(x_n)}{f'(x_n)}$$ is ...
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1answer
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Newton-Raphson Non-Linear Tridiagonal

I have the following problem where I am asked to solve a system of nonlinear equations. I am positive that I have to use Newton-Raphson with the Jacobian. My problem is that I don't fully understand ...
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Apply Newton's Method to approximate the $x$-coordinates of all intersections: $y=1=e^xsin(x);0<x<\pi$

Apply Newton's Method to approximate the $x$-coordinates of all intersections: $$y=1=e^x\sin(x)\qquad 0<x<\pi$$ Let $f(x)=e^x\sin(x)-1$ and $$x_{n+1}=x_n +\frac{1-e^x\sin x_n}{e^x(\cos x_n+\sin ...
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1answer
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Computation of the inverse of a $p$-adic integer in $\mathbb{Z}_p^{*}$

Let $c \in \mathbb{Z}_p^{*}$, i.e. $\left| c \right| = 1$ and define a sequence $(x_i)_{i \geq 0}$ by $x_0 = a\quad$ and $\quad x_{i+1} = 2x_i - cx_i^{2},\quad i = 0,1,2,....$ , where $a$ is any $...
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When does this nonlinear system of equations have a solution?

Consider the following system of equations $$\begin{cases} ||x-k||_p = t\\ f(k+x) = f(k) \end{cases}$$ where $x \in R^n$, $k \in R^n$, $f: R^n \mapsto R$ is a convex function and $|| \cdot ||_p$ is ...
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how to solve $t(t−\sin t)^{5/3}=0.6\pi$

How to solve this question? Using the Newton method? I have tried using this approximation, $\sin t ≈ 120(\pi−t)t/\pi^5$.
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Bisection Method vs Newton's [duplicate]

Why is Newton's root finding method faster than the bisection method? I cant seem to find a good explanation besides calculations of the error through each iteration of each algorithm and comparing ...
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Imbalance of variables in Mixing Newton's method and Linear solver for a Non-linear system

Problem Solving a non-linear system of equations. Number of variable is same as number of equations. When I fix a set variables (say $\vec{y}$) and keep another set free (say $\vec{x}$), the system ...
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Newton-Raphson method issue

I have a quick question. I have a system of residuals in the form of $$R_i= (\sum_{j=1}^n a_{ij}x_j)-b_i$$ where $a_{ij}$ and $b_{i}$ are constants. I am trying to show that applying Newton-Raphson ...
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1answer
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Newton's method to find an update rule to compute $\frac{1}{y}$ given $f(x)=\frac{1}{x}-y$

I have to use Newton's method to derive an update rule for finding a root of the form $\frac{1}{y}$ given a specific $f(x)$, where $f(x)=\frac{1}{x}-y$. From the given, $\frac{1}{y}$ is a valid root ...
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1answer
88 views

Quadratic convergence of a specific iteration (Steffensen's method)

DEFINITION (QUADRATIC CONVERGENCE) : Let $\left\{x_k\right\}$ be a sequence of real numbers and $\xi \in \mathbb{R}$. We say that $x_k \to \xi$ quadratically if and only if \begin{align*} &(i) \...
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1answer
101 views

A convergence problem with Newton-Raphson iteration

I'm stuck with a problem with Newton-Raphson iteration. Below I give definition of the iteration, then I state the problem and provide my incomplete attempt. DEFINITION : Let $f$ be a differentiable ...
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2answers
37 views

Why am I getting no order of magnitude for the error in a Newton-Raphson method problem?

Here is the sequence equation I am given: $$ a_{k+1} = \frac{1}{2}\left( a_k+ \frac{n}{a_k} \right) $$ I am also given $n = 50$ and $a_0 = 7$. This is what I calculated: $$ a_1 = \frac{99}{14}, ...
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0answers
143 views

correctness of Mandelbrot set distance estimation rendering method

I came up with an algorithm that seems to work well in practice for (interior and exterior) distance estimate rendering of the Mandelbrot set, for each starting point $c$: $d := 0$ $z := 0$ $m := \...
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0answers
37 views

Time complexity of a variation of Newton-Raphson method

Recently I was introduced to the Newton-Raphson method for finding roots of a polynomial function. I looked up the proof of it and I found this. I found a variation of the Newton-Raphson method by ...