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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Existence of bounded derivative inverse (Deuflhard Exercise 1.1)

The following exercise is from Deuflhard's "Newton Methods for Nonlinear Problems" : Given a nonlinear $C^1$-mapping $F:X\to Y$ over some domain $D\subset X$ for Banach spaces $X$, $Y$, each endowed ...
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Is it possible to use Newton's method on a function with an integral?

So I have a root solving problem where I want to find x, but my equation is: $$ 0=1-p+c_2\int_0^Xt^{\alpha-1}e^{-t/2}dt $$ where $0\le p\le1, c_2\ge2$, $1\le\alpha$. Is it possible to use Newton's ...
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1answer
38 views

Calculating $\frac{1}{a}$ using Newton-Raphson method

I have a computer that doesn't implement division operation (it has only addition, substraction and multiplication). I need to find a method to find the approximate value of $\frac{1}{a}$, where $a\in ...
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27 views

Finding root of function, possible Lambert function?

So this is my function, and I'm trying to find the root where f(x)=0: $c_1-\frac{2}{c_2}(x+2)e^{-x/2}=0$ where $0< c_1\le1$ and $c_2\ge2$ This is what I got thus far: $c_1c_2-2(x+2)e^{-x/2}=0$ ...
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1answer
36 views

What to use as initial guess for Newton Raphson Method?

The root solving method Newton Raphson converges quickly to the estimated root value but requires a 'close' enough initial guess to converge. I have read that an initial value is often chosen by use ...
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1answer
13 views

Newton method exchanging row

suppose to have a function $F(x,y,z) = [ f_1(x,y,z),f_2(x,y,z),f_3(x,y,z)]$ and that $f_1$ depend only by x, $f_2$ depends only by y and $f_3$ depends only by z. Now if I apply newton method I can ...
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1answer
21 views

Modified Newton method and contraction principle

I am studying Newton's method modified by the book Zorich, Mathematical analysis II, page 39,40: It seems to me, if I make no mistakes, that there is a problem in the derivative of $ A (x) $. The ...
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33 views

Proof Superlinear Convergence Newton's Method

Lets say a sequence $(x^{(k)})_{k\in\mathbb{N}}\subset\mathbb{R}$ with limit $x^\ast\in\mathbb{R}$ converges superlinear if $$\lim_{k\rightarrow \infty}\frac{|x^{(k)}-x^\ast|}{|x^{(k-1)}-x^\ast|}=0.$$ ...
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Newton-Raphson Method and Prove

I have studying numerical analysis and came across this question. Find the approximate value of $\sqrt5$ by using Newton-Raphson method. Take initial approximation as $2$ and $\epsilon = 10^{-6}$. ...
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Broyden method issue in LP optimization with Interior Point method

Good afternoon, I implemented the interior point method trying to follow this article here. Let's recap the the context, I've a problem to optimize with a objective function $f(x)$ a set of ...
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What can be the best initial guess (seed value ) for finding e^x using Newton Raphson?

I'm trying to find out e^x using the Newton Raphson method. At the same time, I want to reduce the number of iterations to compute e^x but for that, a precise initial seed value is required but I'm ...
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1answer
36 views

When is the order of convergence of Newton-Raphson greater than 2?

If some function $f$ has a simple root at $x_*$ then I know that the order of convergence of the Newton-Raphson iteration is at least 2. But when is this order strictly greater than 2?
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Is there general formula for initial guess of Newton-Raphson method for finding roots

A few days ago, we had quiz and I solved 3/4 of questions, and I realized that the 4th question which was mentioned as wrong. Question was ...
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54 views

Find $1/\sqrt{a}$ with the Newton's method

I have sequence defined by : $$x_{n+1} = x_n - \frac{x_n^{-2} - a} {\frac {-2}{x_n^3}}$$ I need to find the sign of $x_{i+1} - x_i$ for $x_i ∈ ]0, \frac {1}{\sqrt{a}}[ $ and for now i only find ...
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2answers
44 views

Global convergence of Newton's method for square roots

To find approximate $\sqrt a$ we can use Newton's method to approximately solve the equation $x^2 − a = 0$ for $x$, starting from some rational $x_0$. Newton's method in general is only locally ...
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30 views

Show that order of convergence of recurrent sequence is at least $3$.

We define the recurrent sequences $(x_n)$ and $(z_n)$ : $$ x_{n+1} = z_{n+1} - \frac{f(z_{n+1})}{f'(x_n)} $$ $$ z_{n+1} = x_{n} - \frac{f(x_{n})}{f'(x_n)} $$ Show that the order of convergence of $(...
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45 views

newton raphson question [closed]

Find a cut point of the $y = x^3-4x-5$ and $y = e^x-4x-5$ curves by selecting the starting point $x_0 = 3$, using the Newton Raphson method with an error of $10^{-3}$ I can't solve the question. Can ...
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1answer
35 views

Convergence of Halley's method at a Root $x = p$ of $f(x)$

Define an iterative method for finding the roots of a function $f(x)$ by $$p_{n+1} = p_n − \frac{f(p_n)f'(p_n)}{f'(p_n)^2 - \frac{1}{2}f(p_n)f''(p_n)}$$ where $f(x)$ is at least twice differentiable....
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Implement the Newton iteration method for multivalued vector functions

I need to implement the Newton Iteration method for multivalued vector functions. I have this differential equation that I have to solve $$\frac{dA}{dt} - \epsilon \frac{3i}{2} \lvert A \rvert^2 A - \...
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PDE and continuation method / Newton's method

I want to solve the nonlinear elliptic equation $$ \begin{cases} \Delta u+ f(u)=g \;\; \text{in} \; \; (0,1)\\ u(0)=u(1)=0 \end{cases} $$ where $f\in C^1(\mathbb{R})$ and $g\in L^2(0,1)$ My idea, ...
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The roots of quintic polynomial

I have some region $[a,b],$ and the polynomial $P(x)$ in it $(\deg P(x)=5)$: Given these conditions $P(a)\cdot P(b)\lt0$ $P'(x)\gt 0$ or $P'(x)\lt 0$ when $x\in (a,b)$ $P'(a) = P'(b) = 0$ I know ...
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Prove if $f^{'}$ and $f{''}$ have opposite signs, then Newton method starting at $x_0$ = a must converge to x∗.

Let $f ∈ C^2 ([a, b]), f(a)f(b) < 0$, $f^{'}$ and $f{''}$ do not vanish and do not change signs on $[a, b]$. Prove if $f^{'}$ and $f{''}$ have opposite signs, then Newton method starting at $x_0$...
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Newton's method on Hilbert space problems / References

Let $S:H \rightarrow K$, a nonlinear functional. Where $H$ and $K$ are two Hilbert spaces. I have the existance of the zero of $S$ i.e $$\exists x^* \in H ,\, \, \, S(x^*)=0_{K}.$$ So in order to ...
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1answer
19 views

zero-finding by Newton Method - multivariate function

I have a function of the type (for simplicity I use a similar and more straightforward function): $ f(x) = \Vert Ax - b \Vert, \quad A\in R^{m\times n}, x \in R^n, \text{ and } b \in R^m$. I would ...
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1answer
33 views

Newton method and machine learning

There is some debate about why Newton method is not widely used in machine learning. Instead, people tend to use gradient descent. Some people claim that Newton method is not used because it involves ...
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1answer
35 views

Newton's method: why do we need $f\in C^2[a,b]$?

My lecture notes give the following derivation of the Newton-Raphson method for estimating solutions of the equation $f(x)=0$: Let $f \in C^2 [a,b]$ and assume that $f$ has a root $x^*\in [a,b]$. Let ...
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32 views

Newton's method derivation and gradient descent.

I am confused by answers here and there... I want to clarify on derivation process of Newton's method. So basically taking first 3 terms of a Taylor expansion: $$f(x+h)=f(x)+h f'(x)+\frac{1}{2} h^2 f'...
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28 views

Newton's method understanding

I want to understand how newton's method is derived from Taylor expansion, and as many answers show that $$f(x+h)=f(x)+h f'(x)+\frac{1}{2} h^2 f''(x)+O\left(h^3\right)$$ and would simply it to : $$f(...
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1answer
38 views

Global convergence for Newton's method in one dimension: number of overshoots

Consider the problem of finding the roots of $f(x)$. We assume that there is a single root $x_*$ between $a$ and $b$, $a < x_* < b$. Assume also that the sign of $f''(x)$ does not change for ...
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1answer
50 views

Newton iteration converges monotonically [closed]

Suppose f is a real function of one variable, f(x)=0 has a solution x*. $f'$ is Lipschitz continuous,$f'(x*)$ is nonsingular. Let $x_0$ is sufficiently close to x* and $f(x_0).f''(x_0)>0$,then the ...
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2answers
72 views

Bounds on the root of $x^{d+1} - x^d - 1$

For integer $d>0$, consider the polynomial $f(x) = x^{d+1} - x^d - 1$. It is easy to see that $f(x)<0$ for $x\in[0,1]$ and $f$ is strictly increasing for $x\geq 1$, so there is one unique ...
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Explanation for condition for newton's method

Newton's method solves for $f(x)=0$ by iterating over possible solutions. the rate of convergence of this scheme is quadratic if 3 conditions are satisfied: Why does $f''$ have to be continuous? ...
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How to find the Interval $[a,b]$ for a contraction mapping for $f(x)=x-\cos x$ where $a,b$ exists in $[0,1]$?

So, using Newton's method, I know the fixed point for the contraction mapping is approximately $0.739$. However, I'm not sure how to go about finding the interval for the contraction mapping. Every ...
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42 views

How to apply Newton's method to this system of two equations?

I'm a bit stuck. We're given the following system of equations: $$3x-2y+1=0$$ $$12x+3y-18=0$$ And they ask us whether Newton's Method converges after only one iteration. It's similar to this question ...
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1answer
48 views

Newton's Method for linearly dependent system of equations

Givens is this system of equation: $$+2x + 5y = -8$$ $$-2x - 5y = 8$$ I'm asked: Whether the system has a unique solutions Whether Newton's Method converges after only one iteration Well, for the ...
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1answer
27 views

Newton's Method convergance

Lets say I have a function $f(x) = x^TAx+b^Tx+c$ Using Newton's method to find a minimum, I get $$x_1 = x_0 + \frac{x^TAx+b^Tx+c}{f'(x)}$$ How would I prove that this converges in one iteration, ...
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Wolfram Alpha is doing something I don't understand, please help

I am using Newton Raphson method to obtain the velocity of a chemical reaction, and I needed to derive the next equation:$$\frac{d}{dk}\left(\left(\frac{0.35}{k}\right)^{\frac{k}{k-0.35}}\right)= -0....
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22 views

Selecting guess matrix for Newton method to calculate inverse of a matrix

I have implemented the newton method as follows to calculate inverse of a matrix(input in my case)and x0 being the initial guess matrix.But the value of the iteration is not stable.I suspect the ...
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1answer
35 views

Newton's Method derivative calculation

I'm trying to reduce this form of Newtons Method for: $f(x) = x^2 - a$, $f'(x) = 2x$ $$x_1 = x_0 - \frac {f(x_0)}{f'(x_0)}$$ to the form, which I am told is the correct expanded form: $(x/2) + (a/(...
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65 views

Newton's Method for Cube Root

I have a text which claims the following (a) is Newton's method for cube roots, where $y$ is an approximation to the cube root of $x$: It's my understanding the form can be derived from: where each $...
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35 views

Newton Method (Root Finding) Theoretical Problem convergence of $x_{n+1} = x_n - \lambda f(x_n)$

$f(x)$ is a function such that for every $x$, $f'(x)$ is well defined and $0<m<f'(x)<M$. We are using a Newton-like method to find its root. consider sequence $x_{n+1} = x_n - \lambda f(x_n)$ ...
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89 views

Finding all positive real roots of $e^x - x -2 = 0$ using Newton Raphson's method.

How should I go approach this problem if I need to solve this problem by hand? Is there a general formula that I can use?
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1answer
113 views

Multivariate Newton-Raphson in R language (equations that contains integral)

I am trying to apply multivariate Newton-Raphson method using R language. However I have encountered some difficulties to define functions which includes the integral in the equations. For instance, ...
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1answer
41 views

Problem about applying the Newton's method to a system

Problem The amount of pressure needed to sink an object in a bed of homogeneous soft soil that lies above a base of hard soil can be estimated using the pressure needed to sink smaller objects into ...
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1answer
27 views

Minimum iterations to guarantee a forward error of $\epsilon$

can anyone shed some light into this for me. I am asked to numerically solve $$x\arctan(x) = 1$$ and to ensure that the error $\epsilon$ is less than $10^{-3}$. Now I usually use Newton's method for ...
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1answer
46 views

Darboux' theorem on the convergence of Newton's method

I've only found the text in French, which I just don't read well enough to understand what is going on. Any pointers to a clear proof in English? The paper is Darboux, "Sur la méthode d'approximation ...
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27 views

Can I compute the first and second partial derivatives along a given direction using the gradient and Hessian?

I have an analytical function $f:\mathbb{R}^n\to\mathbb{R}$ that I want to minimise using Newton's method. Given a point $\mathbf{x}_i\!\in\!\mathbb{R}^n$, I compute the gradient, $\nabla\! f(\mathbf{...
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21 views

How to find $x_1$?

let $f(x) =x^2-5$ for $x \in \mathbb{R}$ . Let $x_0 =1$ . If $\{x_n\}$ denotes the sequence of iterates defined by the newton -raphson method to approximate a solution of $f(x)=0$ . Find $...
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27 views

Convergence of Newton's method when minimizing a strongly convex function

If I have a multivariate function f twice differentiable which is strongly convex, and smooth as well, so $\mathit \nabla ^2f - \mu I$ and $\mathit LI - \nabla ^2 f$ are both positive definite at ...
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I need advice on the use of Gradient descent vs. Newton's method to minimise a multi-variable objective function.

I have a function $f:\mathbb{R}^n\to\mathbb{R}$ with an analytical expression. The function itself is very complicated, so I will omit the definition for brevity. It is a highly non-linear function ...

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