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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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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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24 views

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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6 views

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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23 views

Convergence of damped Newton’s method [on hold]

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
15 views

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
26 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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1answer
217 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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8 views

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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12 views

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
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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
18 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
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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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13 views

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
46 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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32 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)...
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27 views

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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26 views

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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35 views

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
27 views

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
54 views

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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0answers
26 views

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
53 views

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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2answers
36 views

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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18 views

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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0answers
24 views

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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38 views

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
60 views

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 ...
3
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1answer
72 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
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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
34 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
128 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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34 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 ...
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26 views

How can I understand Newton's root finding algorithm from the equation?

How can we prove Newton's method works correctly? Here, let me describe the equation by $f(x)=\frac{1}{x}-a$ for some $a$. The root of $f(x)$ is $\frac{1}{a}$. Its derivative is $f'(x)=-\frac{1}...
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29 views

Solving non linear equations

I want a int of how to set the resolution (using Newton Raphson method) in order to solve this equation : $$ k_1 = f(t_i+ \frac{1}{2} h, y_1 + \frac{1}{2}k1) $$ I'm using python but I want ...
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1answer
36 views

Show that the equation $y=e^x/x^3$ has a root between $1.2$ and $1.3$ [closed]

Show that the equation $y=e^x/x^3$ has a root between $1.2$ and $1.3$ Hey, having trouble with this one. Would appreciate a hand, not sure how to go about the exponent and such.
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64 views

Confusion in Second Derivative of a Vector Valued Function

I have a slight confusion in Newton Raphson method applied in Higher Dimensions. I have a vector valued function $f(x)$ which maps from $\mathbb{C^{m}}\rightarrow \mathbb{C^{m}}$ whose root I have to ...
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2answers
73 views

How to solve equations like $\alpha \sin x -\beta\sin 2x +\gamma=0 $

Can I solve this equation without Newton-Raphson method? I have $\alpha=47.02$ $\beta=112.5$ and $\gamma=50$. When I have to use Newton-Rapson to solve trigonometric equations ? I will greatly ...
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2answers
186 views

Newton's method for a vector field

Let $f : \mathbb{R}^n \to \mathbb{R}^n$ be $C^2$ and let $f(x^*)=0$. Since $$f(x^*) \approx f(x) + Df(x) (x^* - x)$$ we can have the iterative procedure $$x_{k+1} = x_k - Df(x_k)^{-1} f(x_k)$$ Is ...
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58 views

Solving for an implicit function numerically?

I am trying to come up with a systematic way for solving an implicit "functional" equation. That is, given $$g(x) = f(x + a \cdot f(x))$$ how would one (numerically) recover the functional form of $...
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1answer
58 views

Prove that a specific fixed point iteration is locally convergent [duplicate]

Let $g : I \rightarrow \mathbb{R}$ be a $C^1$ map such that $g'(x) \ne 0$ for any $x$ in $I$. Assume that there exists $r \in I$ such that $g(r)=0$. Prove that for $\eta \in I$ sufficiently close to $...
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2answers
108 views

Newton's method - weird triple oscillation $x^5-x+1$

I'm investigating iterations using Newton's method for $f(x)=x^5-x+1$. I'm getting unusual results though. I've found that some starting values will result with a "triple oscillation" of results ...
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2answers
57 views

Newton-Raphson problem

Consider the equation $\cos x = c$, where $c$ is constant. Find the value of $c$ so that the approximate solution by Newton-Raphson method at the $k$th iteration is $x_k = (-1)^k$, where $k = 0, 1, 2, ...
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0answers
53 views

Root finding method

My prof. mentioned a root finding method called something like Alekhine's method or similar, but I can't seem to find anything about it on google, except for some chess variation. Can anyone help out? ...
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0answers
81 views

Convergence of Newton method and machine precision

I implemented the Newton method to find the non-zero root of $f(x) = 1-bx-e^{-x}$ in Excel and I have tested it for various values of $0<b<1$. However, what I am seeing for some values of $b$ (e....
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1answer
34 views

Quadratic function optimum not aligning with implementation.

Let's define a quadratic function ($x$, $x_0$ and $g$ are vectors and $H$ is a symmetric matrix so, $H^T=H$): $$f(x) = \frac{(x-x_0)^TH(x-x_0)}{2}+g^T(x-x_0)$$ $$=>f(x)=\frac{x^THx}{2}+\frac{x_0^...
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0answers
45 views

Newton's method for fraction

Let $a\gt0$. Start from a convenient equation and use Newton's method to deduce a method to approximate $\frac{1}{\sqrt{a}}$ without divisions. How is the starting value chosen? What is the ...
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0answers
40 views

How to find the minimum of a noisy function that has very narrow valleys?

I am trying to find a set of optimum parameters of my model that provide the best fit. I chose to make this into an optimization problem where I minimize my $\chi^2$ per degree of freedom and try to ...
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0answers
82 views

If fixed-point iteration has linear convergence, how can Newton's Method have quadratic convergence?

Newton's Method for finding the roots of a function can be considered a type of fixed point iteration of $g(x) = x - \frac{f(x)}{f'(x)}$, since $f(k) = 0 \rightarrow g(k) = k$. But it is well-known ...