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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Let $f:\mathbb{R}\rightarrow \mathbb{R}$ with $f(x) = xm, m>1$ . Write Newton’s method for approximating root $x^∗ = 0$ of $f$ starting with $x_0= 0$. [closed]

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ with $f(x) = xm, m > 1$ Write down Newton’s method for approximating the root $x^∗ = 0$ of $f$ starting with initial guess $x_0= 0$. Express the $n^{th}$ ...
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What is the order of a linear multistep method that uses Newton-Raphson?

I know that the order of convergence of Newton's method is quadratic, but I don't know how that translates into the resulting method.
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BFGS computational complexity derivation

The update for the Hessian using BFGS is given by: $$H_{k+1}=(I-\rho_ks_ky_k^T)H_k(I-\rho_k y_ks_k^T)+\rho_ks_k s_k^T$$ where $\rho_k=\dfrac{1}{y_k^Ts_k}$. Nocedal and Wright, Numerical Optimization ...
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Newton's method is Quasi-Newton when the function is a non-degenerate quadratic

With the update step $\textbf x_{t+1} = \textbf x_t - H_t^{-1}\nabla f(\textbf x_t)$, where $H_t \in \mathbb R^{d\times d}$ is symmetric and satisfies $\nabla f(\textbf x_t) - \nabla f(\textbf x_{t-...
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How do I solve this non-linear differential equation?

I have a formula describing a non-linear system which is as follows: $-y(x)'' + A|y(x)|^2y(x) = k^2 y(x)$ where A is a constant that I can choose to be a positive or negative integer. I also have ...
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Bad Convergence, Gradient Descent

I'm trying a Gradient Descent, (maybe Newtons Method?) for Linear Regression and getting wildly different solutions from the faster, more straight-forward linear equations, but can't find my mistake ...
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What is the Matlab code for Newton Raphson method to solve the polynomial system?

I want to approximate the zeros of the following system of polynomial equations using the Newton-Raphson method: \begin{align} f(x,y) &= x + \frac13 y^9 + \frac19 x^{243} + \frac{1}{27} y^{2187} =...
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  • 9,587
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How to solve the system of nonlinear equations in N-R method or other numerical methods?

Consider the system of infinite series \begin{align} &F=x+\frac{y^{3^2}}{3}+\frac{x^{3^5}}{3^2}+\frac{y^{3^7}}{3^3}+\frac{x^{3^{10}}}{3^4}+\frac{y^{3^{12}}}{3^5}+\cdots=0 \\ &G=y+\frac{x^{3^3}}...
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Does Newton Method with Armijo rule find solution for quadratic in one step?

Consider a quadratic $f(x)=\frac{1}{2}x^TAx-c^Tx$, with $A \in \mathbb{R}^{n \times n}$ positive definite and the Newton's method with Armijo search applied to minimize $f$ over $\mathbb{R}^n$. It is ...
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Can you please help me whether $N-R$ method or other methods converges?

Consider the system of equations \begin{align} &f(x,y)=x+\frac{y^4}{2}+\frac{x^{32}}{4}+\frac{y^{128}}{8}=0 \\ &g(x,y)=y+\frac{x^8}{2}+\frac{y^{32}}{4}+\frac{x^{256}}{8}=0. \end{align} I want ...
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How can I apply Smale's alpha theory to determine if an initial guess $x_0$ converges?

I recently encountered a book called "Complexity and Real Computation" where Smale's alpha theory on the Newton method is explained. I tried doing some of the exercises but was unable to ...
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For which starting points for $f(x) := \sqrt{1+x^2}$ does Newtons Method converge?

I am stuck at the following exercise: For which starting points for $f(x) := \sqrt{1+x^2}$ does Newtons Method converge? I recently learned about Newton's Method in class and the real theorem ...
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Does the newton method with $f(x)=x^p$ converge superlinearly?

Let $f(x)=x^p$ with $p>2$. The steps for the newton method are known as $x_{k+1}=x_k+d_k$ and $d_k = -\frac{\nabla f(x_k)}{\nabla^2 f(x_k)}$ I want to check if the Newton method converges ...
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Newton's Method for Nonlinear System with Constraints

I have a local solution of a dynamical system $\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x})$: \begin{equation} \mathbf{x}(t) = \mathbf{g}(t;\mathbf{A}), \end{equation} where $\bf{f},\bf{g}:\mathbb{R}^n\...
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Convergence condition Newton method for quadratic system of equations

So suppose I want to solve a system of $n$ quadratic equations given by $f: \mathbb{R}^n \rightarrow \mathbb{R}^n $ using the Newton method, $x_{n+1} = x_n - J^{-1}(x_n)f(x_n)$. Do there exist general ...
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Why does the pure Newton method use step size $\alpha = 1$?

We know that pure Newton method is derived from the Taylor second order expansion. Its step size equals to $1$. Also there exists Newton method with step search. Here are my questions. Does the ...
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Why does the Newton-Raphson method sometimes not converge?

I'm talking about the Newton-Raphson method for finding square roots of matrices: $$\begin{cases} X_{k+1} &= \frac{1}{2}\big(X_k + X_k^{-1}A\big)\\ X_0 &= A \end{cases}$$ as a "...
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Root finding and automatic differentiation

Consider the equation $z = f (z, x)$. We would like to find $z^{\star}$ for $f$ such that $z^{\star} = f (z^{\star}, x)$. One way to do this problem is through naive iteration: $z^{(k + 1)} = f (z^{(k)...
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1 answer
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Initial vector for Newton-Rhapson method for several variables

I've been worked in a exercise about Newton's method on several variables, more specific the exercise 12, chapter 10.2 of Burden-Numerical Analysis pag. 646 https://faculty.ksu.edu.sa/sites/default/...
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Show that the equation has exactly one solution on $[1, 2]$,then find the solution correct to $4$ decimal places using Newton’s method

Show that the equation $𝑥^3 + 2𝑥 = 5$ has exactly one solution on $[1, 2]$. Then find the solution correct to $4$ decimal places using Newton’s method. Looking at the problem, I wonder how do I ...
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Is Newton's method a descending algorithm

Consider the following optimal problem $$ \min_{\mathbf{x}} f(\mathbf{x}) $$ where $f$ is continuously differentiable and positive definite. The Newton iterative scheme is $$ \mathbf{x}_{k+1}=\mathbf{...
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Newton Raphson Method

Problem Statement: There were 46 crude oil spills of at least 1000 barrels from tankers in U.S. waters during 1974-1999. Data: Ni = the number of spills in the ith year bi1 = the estimated amount of ...
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2 votes
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approximate the solution $x=2$ using newton's method for $P(x)=-1536+6272x-11328x^2+11872x^3-7952x^4+3528x^5-1036x^6+194x^7-21x^8+x^9$

I need help with this excercise. I know that $x=2$ is a solution for $$P(x)=-1536+6272x-11328x^2+11872x^3-7952x^4+3528x^5-1036x^6+194x^7-21x^8+x^9$$ I want aproximate the solution $x=2$ using newton ...
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NR to find the maximum value

We have a reinforced concrete beam asymetric section. Strain of the section is $e=a z+b y+c$ because sections remain planar. Axial force and bending moments are $N(a,b,c)$, $M_y(a,b,c)$ and $M_z(a,b,c)...
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Proving convergence of Newton-Raphson using contraction mapping theorem

I'm trying to prove there exists an $\epsilon \in \mathbb{R}$ such that for a root, $p$, of a function $f \in C^2$, we have that $\forall q \in (p - \epsilon, p + \epsilon)$, $q$ converges to $p$ due ...
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Not reaching to the correct answer using Newton Raphson Method

I am trying to find the inverse of the following function using Newton Raphson Method: (This is the function) by transforming it into this. Here, k is a constant, and we know the value of y hence it ...
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How to use Newton-Raphson method to check differentiability of a function?

Let's say we have a function f(x) = x^a + x^b + x^c (a, b, c, x ∈ R) How can we check the differentiability of f(x) first using the Newton-Raphson's method? Using ...
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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|{...
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What is the Jacobian's form of a discretized PDE?

I need to solve a PDE using finite difference method with the Newton-Raphson Method. In the merhod, I need to calculate the Jacobian matrix of the residual of the discretised PDE. The PDE may be like: ...
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1 answer
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How I find $\lim_{n\to \infty} \frac{x_n}{\sqrt{n}}$ using $x_n=x_{n-1}+\frac{m}{x_{n-1}}$? [duplicate]

Let $(n,m)\in \mathbb{N^{+}}$ and $x_0=p$ such that $p^2>2m $ , We have the following rational dynamical system $$x_n=x_{n-1}+\frac{m}{x_{n-1}}$$ I would like to find $$\lim_{n\to \infty} \frac{x_n}...
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2 votes
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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 ...
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1 vote
1 answer
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Newton's method in higher dimensions

To calculate the inverse of a quadratic matrix A, we could solve the equation $F(X):=X^{-1}-A=0$. I need to show that if X is invertable, then $DF(X)(\Delta X)=-X^{-1}\Delta XX^{-1}$ where DF(X) is ...
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For what start value does Newton's method converge?

I have the equation $f(x)=1/x-a=0$ and I need to find necessary and sufficient conditions for the start value $x_0$ so that the method converges. First I set up the sequence $x_{k+1}=x_k-\frac{f(x_k)}{...
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Newton-Raphson method convergence criteria for a system of two equation with two unknowns

I am having a system of two non-linear equations and I want to compute the root. For example the system is $$f_1(x,y)=0 \tag{1}$$ $$f_2(x,y)=0.\tag{2}$$ In this post, it is shown that the root of the ...
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Difference between Newton's method and Modified Newton's method?

$X_{k+1}=X_k-H(X_k)^{-1}*G(X_k)$ This is what I know to be a newton Raphson method. However, one of my hw question is asking about Modified Newton's method, and what we have learned in class is Quasi ...
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Prove that Newton's method converges for x^2-p=0

Given the equation $x^2-p=0, p>0$, one has to show that Newton's method will always converge for every initial value $x_0>0$. I have found the sequence $x_{k+1}=x_k-\frac{f(x_k)}{f'(x_k)}=x_k-\...
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1 vote
1 answer
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Tune an exponential regression estimate using calculus

I have an exponential regression equation that I use to predict the future condition of roads: $$y=a+be^{cx}$$ Modified for my purposes as $$y=21-e^{a x}$$ Using the normal equation $$a=\frac{\sum_{i=...
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Finding the roots of an absolute value natural log function, regarding the usage of Newton-Raphson

This is an exercise I saw in my last test, I've been practicing it but I'm currently a bit lost when finding the roots, here's the exercise: Given $f:f(x)=-x^2+2x+3$, be F/F(x) is a primitive of f ...
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Need specific example where Newton's method doesn't converge [duplicate]

Newton's method has a local proof of convergence, however it may not converge in some cases. I'm trying to find a case where it has more than one limit point, some behavior kinda like the function on ...
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3 votes
3 answers
261 views

Why does the sign in Newton's method matter?

Deriving Newtons Method visually as with the help of a right triangle and assuming $x_1$ lies the left of $x_0$ we get $$x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}$$ Using slope over run. but if we assume $...
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Root Finding: Use Newton Method in given Interval (or alternative?)

I need to find a root of an equation in a given interval. The Problem is that there are more than one roots. But only within [0,$\pi$] I'm interested in the solution. So I implemented a function ...
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Do I apply the univariate or the multivariate version of the Newton-Raphson iterative method to this equation?

The equation in question is $f(x) = x^4 - 4x - 2$ and I want to minimise the function. Regardless of whether it is ideal or not to use Newton-Raphson, which one would I use? Sorry if this is a silly ...
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Is $f'>0$ enough for Newton's method to converge? [duplicate]

Suppose $f: \mathbb R \to \mathbb R$ is smooth, that it has a root and that $f'(x)>0$ for all $x$. Does this guarantee that Newton's method will converge (quadratically) for all choices of initial ...
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Is this Fréchet derivative really a bounded functional?

I am currently studying Newton's method on Banach spaces, specifically as it pertains to the linearization of nonlinear PDEs. In a sample problem, the following operator between Banach spaces came up: ...
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Show that Newton update in two variables converges

I am having two update rules of a two variable function $f(x, y)$, i.e., $$x \mapsto h_1(x,y):=x - \frac{f(x, y)}{\partial_x f(x,y)} \tag{1}$$ $$y \mapsto h_2(x,y):=y - \frac{f(x, y)}{\partial_y f(x,y)...
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Familiar with splitting variables in 'two-step' Newton-Rapshon method?

Introduction Let $f(x,y,z)$ be a continuous function, which I want to optimize using Newton-Rapshon method by finding the roots of the gradient, $\nabla f$. $f$ is convex in $x$ and $y$, but whether $...
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2 votes
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Taylor approximation of $\phi_1(\lambda) = \frac{1}{\sqrt{\psi(\lambda)}} - \frac{\sigma}{\lambda}$

I am reading this paper. In some point of the analysis the non linear equation $$\phi_1(\lambda) = \frac{1}{\sqrt{\psi(\lambda)}} - \frac{\sigma}{\lambda} \tag{A}$$ is studied, i.e. $(6.7)$ in the ...
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I want to prove that the Newton's method is of "order 1" if the derivative is 0 at the root.

Suppose f is of class C^3. That is, f is three-times differentiable and f''' is continuous. Let c be a zero of f. In general, The Newton's method is of order 2. But I want to prove this statement: If ...
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Time Complexity of this unconstrained optimization problem

Following objective is to be maximised: $\sum_{i=1}^n c_i.x + \sum_{j=1}^{300} e^{-x_j^2}$ where $x \in R^{300}, c_i \in R^{300}$ and $x_j$ is the $j^{th}$ element of the vector $x$ We need to find ...
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Newton-Kantorovich theorem insufficient when the Jacobian is singular at solution

I'm trying to improve my understanding of why the Newton-Kantorovich theorem is, in some ways, insufficient in the singular case. Let $f:\mathbb{R}^n\to\mathbb{R}^n$, and let $Df$ denote its (Frechet) ...
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