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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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Proof of Newton-Kantorovich theorem, Wikipedia version

Context I have recently been researching the Newton-Kantorovich theorem after wondering about convergence criteria for the Newton-Raphson method in numerical analysis, as it seems to be the most ...
Victor Liu's user avatar
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How does extra equations affect Newton-Raphson method's performance on solving system of non-linear equations? [closed]

I'm working with model updating, in which the model's parameters are adjusted in order to reduce its ouput error in relation to a reference. For this, I would like to compare minimizing a single ...
Marcus Vinícius Medeiros's user avatar
-3 votes
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Use Newton Method and Generalized Newton Method to Solve the following problem with necessary process [closed]

Solve the following problem of Numerical Analysis using Newton-Raphson Method and Generalized Newton-Raphson Method correct to $0.0001$ with initial guess $X_{0} = ...
acharyabibash's user avatar
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local superlinear convergence of Newton's method for C^1 functions

i am trying to prove, that there exists an $\epsilon>0$, s.t. for $f\in C^1([a,b])$ with $f(x_*)=0,\;f'(x_*)\neq0$ Newton's method converges superlinear for every starting point $x_0\in[x_*-\...
mappingmoe's user avatar
2 votes
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73 views

Newton-Raphson Method's Convergence

I have a function with three real roots, for which I have to prove the following: There are three intervals in which, for every initial guess, N-R converges to the root. I have this Theorem from a ...
Francisco Javier Maciel Hennin's user avatar
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18 views

How can I derive equation 2.23(Newton-Raphson for Entropy) in NASA CEA analysis

It might sound a bit basic, but, I'd like to follow NASA CEA report I. analysis from the beginning. So, I have to derive the Newton-Raphson equation from the entropy equation, which is one of the ...
dave's user avatar
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1 answer
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Newton-Raphson algorithm proof: derivating a matrix

A particularly useful algorithm when you want to find the zero of a function is the Newton-Raphson method. For simplicity, we begin by examining the simplest case. Given a function and his root $x^\...
user3204810's user avatar
3 votes
1 answer
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Problem with Newton's method (numerical analysis)

I am not understanding how to proceed with this exercise, which asks me to solve $f(x) = 0$ by using Newton's method. It asks me to study the convergence of the sequences $x_k$ (built with Newton's ...
Heidegger's user avatar
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1 vote
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Newton raphson method in multidimension space

I have a question, and that is, lets say $e(w) = x - x* = w^T\phi - x^* \in \mathbb R^n$, where $w \in \mathbb R^{N\times n}$ and $\phi \in \mathbb R^N$ I wish to use Newton Raphson method to find the ...
dead_space's user avatar
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Scaling nonlinear system for iterative numerical solver

In Numerical Recipes (C++) there is a Globally convergent Newton Method that can be used to solve systems of nonlinear equations. For context this is section 9.7 (page 477) of the 3rd edition. ...
RedPen's user avatar
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Is there componentwise descent property of Newton's method

Consider Newton's method for minimizing $f:\mathbb{R}^2 \rightarrow \mathbb{R}$, which is the basically applying it for solving $\nabla f(X)=0$, where $X=(x,y)\in \mathbb{R}^2$. The iteration of ...
karlabos's user avatar
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1 answer
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Choosing the initial seed point in Newton Raphson method

In Newton-Raphson method to find approximate roots I noticed 2 approaches being followed to choose initial seed points for a given equation f(x) and I am confused when to apply which method. First ...
JOYDEEP_MALLICK's user avatar
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Does the accuracy of the iterations of the Newton method transfer to parts of the underlying non-linear equation system?

I'm just wondering one thing. Suppose I have a non-linear system of equations $F(z) = z - d(z) = 0$ for $F: \mathbb{R}^{n} \longrightarrow \mathbb{R}^{n}$. If I apply a Newton method with respect to $...
Donnie's user avatar
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Tangent definition with a newton raphson question example

I'm very confused about the definition of a tangent, I was told its a straight line that touches a curve at a point, but if extended does not cross the curve at any other point. In this question if u ...
j jose's user avatar
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3 votes
1 answer
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Newton root finding and preserving automatic differentiation.

I am applying Newton's root solving algorithm. Suppose the example problem below, solving for $g$ with fixed parameter, $s$: $$ f(g;s) = g^2 - s = 0, \qquad g_{i+1}=g_i - \frac{f(g_i;s)}{\frac{df}{dg}(...
Attack68's user avatar
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1 answer
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Inexact Newton Method convergence

Let $f:\mathbb{R^n} \rightarrow \mathbb{R^n}$ be differentiable and $x^*$ be a root of $f$; $f'(x^*)$ be regular and $f'$ is Lipschitz-continuous in a neighborhood of $x^*$. I have the following ...
J3ck_Budl7y's user avatar
3 votes
1 answer
54 views

How to choose initial (x0,y0) point to approximate a solution of a system of non-linear equations using newton method?

I'm studying Newton's method for solving a system of non-linear equations. A system of non-linear equations, whose (one of the) solutions is ($x^*$, $y^*$): \begin{equation*} \left\{ \begin{alignedat}{...
imensy dy gordy's user avatar
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Newton's methods, do we use Hessian matrix or its transpose?

Im working with Newton method, and during I was trying to see why we got this expression, I fall into a problem, I find the same expression but instead of Hessian matrix I find his transpose. I can't ...
Cauchy_Chlasse's user avatar
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Interval Newton Method - not converging

I need to solve polynomial equation and I know an interval with a root. I try to use Newton method, but sometimes it fails due to overshooting. So instead I try to adopt an interval method, to keep ...
Ilya Kondratev's user avatar
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algorithm guaranteed to converge for convex function

I have a multi-variate convex function, and I want to find its global minimum. We know there is one and only one minimum. Is there any algorithm which is guaranteed to converge to the minimum starting ...
poisson's user avatar
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Solving a quadratic, non-linear non-homogenous recurrence relation (Heron's algorithm for finding square roots.)

Consider the polynomial $f(x)$ such that $f(x) = x^2 - a$ for some constant $a$. Using Newton-Raphson, we can approximate the roots of this polynomial, which will be $\pm \sqrt{a}$, as follows: $$x_{n+...
Kraken's user avatar
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1 answer
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Sensitivity of Newton's method initial vlaue

Can someone explain to me what sensitivty refers to when it comes to root finding? When we say it is sensitive to initial value for x, is because there are multiple roots, or because there might be a ...
Need_MathHelp's user avatar
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0 answers
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Newton's convergence for systems of equation analysis

So I solved a question using Newton's Method for systems of equations. Then they asked: How can you ensure that Newton's method converges as it should? What convergence rate do you observe? My idea ...
Need_MathHelp's user avatar
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0 answers
58 views

Fixed Point for Newton's Method

If a polynomial function has at least one real root, will Newton's Method always converge to one of those real roots? (no attracting fixed point). Is there a counterexample where the guess does NOT ...
MaximeJaccon's user avatar
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1 answer
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How to deduct Newton's method heuristically

Let $\alpha$ be a fixed point of a contractive function $g$ defined in $[a, b] \subset \mathbb{R}$ such that $g'(\alpha) = 0$ and $g''(\alpha) \neq 0$. My first question is how to prove that $x_{k + 1}...
Cyclotomic Manolo's user avatar
3 votes
1 answer
71 views

Show that the recursive sequence $x_{k+1} = |x_k - \frac{x_k}{1-2M^2x_k^2}|$ is monotone

I'm doing some exercises for an upcoming exam, and as part of a larger problem, I want to show that the given recursive sequence: $$x_{k+1} = \left|x_k - \frac{x_k}{1-2M^2x_k^2}\right|$$ is monotonly ...
maibrl's user avatar
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1 answer
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Connection between gradient descent and Newton's method

Given a function $f:\mathbb{R}^d\to \mathbb{R}$, suppose we want to find the minimum of $f$. The gradient descent method finds $x$ that attains the minimum by iterating the following formula: $x_{n+1} ...
Kaira's user avatar
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1 vote
1 answer
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Newton's method - does it always work for functions that are close to the identity function? [closed]

Since Newton's method for finding roots trivially works for the identity function $t \mapsto t$, I was wondering whether it also works for every continuously differentiable functions that is ...
jenda358's user avatar
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0 answers
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For a differentiable function $f$ , does this mean value theorem iteration method converge to the root of $f?$

Suppose $a<b,\quad f:(a,b)\to\mathbb{R}$ is a differentiable, nowhere-linear function$,\ f(c)=0 $ for some $c\in (a,b),\quad$ and $f'(x)\neq 0\ \forall\ x\in (a,b).\ $ Let $x_0=a;\ x_1=b.$ Nowhere-...
Adam Rubinson's user avatar
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0 answers
81 views

Infinite series expansion for roots

Newton's method states that, if there is a root $r$ of $f(x)$ that we want to calculate and $r_0$ is an approximation, then a better approximation is $$r_1=r_0-\frac{f(r_0)}{f'(r_0)}$$We could ...
Kamal Saleh's user avatar
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0 answers
43 views

Newton's method when $f'(\overline{X}) = 0$

Everywhere I read a proof of Newton's method (that one with $x_{n+1} = x_n-f(x)/f'(x)$), it is used that the derivative of the function does not vanish at the root we are looking for. There is a post ...
Vitor's user avatar
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Simple modification of Newton's method problem

Since you can transform Newton's recurrence relation to the original equation $f(x)=0$ by considering the limiting case as $n$ tends to infinity, it's not surprising you can also do it the other way ...
milin's user avatar
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1 vote
0 answers
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variant of Levenberg-Marquardt suitable for Lagrange-Newton method

If a Newton descent is applied to some scalar function of a vector, the Hessian is positive definite in the vicinity of a minimum, but can become indefinite in larger distance from the minimum. This ...
Ralf's user avatar
  • 303
0 votes
1 answer
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A proof that Newton method converges

I am asked to write down the Taylor series for a function $f$ evaluated at $x + h$ in terms of $f(x)$ and its derivatives evaluated at $x$. Then, to use this result to show that if $x_0$ is an ...
sam wolfe's user avatar
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0 votes
1 answer
93 views

Solving 5 equations with 5 variable using Newton Raphson method

I have 5 equations with 5 variables $X_1$, $X_2$, $X_3$, $X_4$, and $X_5$, namely \begin{align} a_{11}X_1 + a_{12}X_2 + a_{13}X_3 + a_{14}X_4 \sin X_5 &= b_1,\\ a_{21}X_1 + a_{22}X_2 + a_{23}X_3 + ...
Mon's user avatar
  • 37
0 votes
1 answer
122 views

Finding Newton method order of convergence

I'm trying to determine how you find the order of convergence of newton's method. I have the formula $$\frac {|x^*-x_{n+1}|}{ |x^*-x_n|^q} = \alpha$$ I'm setting $q=2$ to test for quadratic ...
blov's user avatar
  • 13
0 votes
0 answers
36 views

How to iterate $(x-1)/(x\ln{x})=a$ using Newton Raphson

I am trying to iterate $\frac{x-1}{x\ln{x}} - a= 0$ to solve for $x$ using Newton Raphson, but it is blowing up. Any ideas why? Is this a strange function to find a root for? $$x_{i+1} = x_i - f(x_i)/...
forgettable987's user avatar
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0 answers
64 views

Calculating convergence order for Newtons method

I am trying to teach myself numerical analysis and I am trying to find out how you determine the order of convergence for newtons method. On the wiki there are multiple formulas for newton methods ...
blov's user avatar
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0 votes
4 answers
239 views

How to determine the almost accurate initial guess to calculate the $\ln(x)$ via Newton-Raphson method?

I'm developing my own Real Number class, and for that, I have to implement the Natural Logarithm function. I've used Maclaurin series of Natural log with iteration of 300 or 500. The division method ...
Debtanu Gupta's user avatar
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0 answers
20 views

Order of convergence/convergence rate of variant of Newton's Method [duplicate]

$x_{n+1}=x_n - \frac{f(x_n)}{f'(x_n)}-\frac{1}{2}\frac{f(x_n)^2 f''(x_n)}{f'(x_n)^3}$ For obtaining the formula above, I did the following: $f(x_n+ \Delta x_n) \approx f(x_n) + f'(x_n)\Delta x_n + \...
J P's user avatar
  • 343
0 votes
1 answer
171 views

Is there a self-correcting iterative method for approximating pi without using transcendental functions?

The Newton-Raphson method is an iterative method for finding a root of a function, and it is self-correcting in the sense that any error in the initial input is reduced with each iteration so that it ...
noumenon28's user avatar
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0 answers
59 views

Aitken's $\Delta^2$ Method applied to the Newton's Method

Could Aitken's $\Delta^2$ Method be used to accelerate the convergence of the Newton's Method? I simplified $p_n$=$x_n$-$\frac{(x_{n+1}-x_n)^2}{x_{n+2}-2x_{n+1}+x_n}$ and obtained $p_n$=$x_n-$$\frac{(\...
J P's user avatar
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2 votes
0 answers
61 views

Newton's Method convergence for all approximations in $[a,b]$

Given a function $F(x)$ defined on $[a,b]$ such that: $f$ $\in$ $C^2([a,b])$ $F(a)F(b)$<0 $F'(x) \neq 0$, $\forall$ $x$ $\in$ $[a,b]$ $F''(x)$ $\geq$ $0$ or $F''(x)$ $\leq$ $0$, $\forall$ $x$ $\in$...
J P's user avatar
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0 votes
4 answers
120 views

Why does taking the tangent line improve the approximation in Newton's method?

I have gained a comprehension of the operational process through the discussion located at Why does Newton's method work?. Nevertheless, there is one aspect that remains unclear to me. To initiate,...
Nijat Hamidov's user avatar
3 votes
1 answer
313 views

Newton method with high multiplicity: rigorous proof

I am looking for a rigorous yet reasonably short proof of the following statement: let $f\in \mathcal{C}^{m}([a,b])$, and let $\alpha \in (a,b)$ such that $$ 0 = f(\alpha) = f'(\alpha) = f''(\alpha) = ...
Federico Poloni's user avatar
1 vote
0 answers
92 views

Vector equation where each element is equal to a quadratic form

Is there a way to solve an equation of the following form: $$x_i = {\bf x}^T{\bf A}_i{\bf x} + b_i$$ where $x_i$ is the ith element of vector $\bf x$. I'm guessing this could be equivalently expressed ...
J. Zeitouni's user avatar
0 votes
0 answers
101 views

Excel solver in newton-raphson

I have an excel worksheet that solves system of PDEs using newton-raphson method. The solution obviously depends on some variables that I input. There is a VBA macro (lets call it VBA1) that is used ...
Nurlan Z's user avatar
1 vote
1 answer
100 views

How to find rational power of two using Newton's method?

I know how to find integer powers of two, $2^x = \prod_{i=1}^x 2$, I have memorized powers of two up to 32nd power of two, and I use bit-shifts to calculate them. For integer powers of other numbers I ...
Ξένη Γήινος's user avatar
1 vote
1 answer
193 views

How to find log2 of a number with Newton's method?

I am learning C++ and I find pow, log, log2, exp ...
Ξένη Γήινος's user avatar
0 votes
0 answers
100 views

Algorithm for non-linear system of equations

I would like some tips in figuring out a good algorithm to find the solution of the following system. Let $\theta$ be a constant in $(0,1)$, let $i,l=1,...,N$, let $a_{l}$ and $b_{i,l}$ be some ...
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