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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32 views

How to show that errors exhibit a quadratic convergence

Given the simple cubic function $$f(x) = x^3 - 1 = 0,$$ You can use the Newton Raphson method to solve and approximate one solution to the function like so: n $x_n$ Absolute Error 1 1.41667 0.41667 ...
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For a strongly convex function, does the initialisation of $x_0$ matter?

So I am curious what the convergence conditions are for a strongly convex function using the Newton Raphson method. If the function is strongly convex then if the First Order Necessary Conditions hold,...
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Numerical Methods basic question [closed]

Kindly give the solution for the given question
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87 views

approximate value of $\sqrt7$

the objective of the exercise that I am trying to solve is to find the approximate value of $\sqrt7$,and to do that I should solve numerically the equation $f(x) = x^2 -7 =0$ , the exercise have ...
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1answer
65 views

Solve transcendental equation: At $α\ll 1$, $\left|\cos x + \alpha \frac{\sin x}{x}\right| > 1$, determine the width of of the $k$-th zone at $k\gg1$.

To solve this transcendental equations approximately: At $\alpha \ll 1$ find the positive solution of inequality: $\left|\cos x + \alpha \frac{\sin x}{x}\right| > 1$, they are divided into series ...
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30 views

Analytical solution for a difficult nonlinear PDE

Is it possible to compute the analytical solution for this nonlinear pde? It doesn't seems to work with Sympy but it doesn't i can do it with it. The point is to prove that the convergeance order of ...
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27 views

Is it possible to calculate the feed forward Hessian inverse?

Did I made a calculation error? Say we had a simple one layer perceptron where: $f$ is the activation function, $w$ is the weights matrix, $b$ is the bias vector, $x$ is the input vector, $y$ is the ...
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1answer
21 views

Linear Function and Newton's Method

I want to show that if a function is linear for example F(x) = 3x + 3, then after one iteration of Newton's method I can find the x-value such that F(x) = 0. In this case, after one iteration x = -1. ...
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27 views

Newton's method for nonlinear equations example

This is a problem from the book "Numerical Methods for Unconstrained Optimization and Nonlinear Equations" by J. E. Dennis, Jr., Robert B. Schnabel. For each of the functions $f_1(x)=x, f_2(...
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1answer
51 views

Taylor Series, Newtons Method and Newton Raphson

I am currently learning about different optimization methods. That is, I am interested in finding the maxima/minima of functions. The last method I have studied is Newton Raphson. I understand Newton ...
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1answer
46 views

What does it mean when Newton's method encounters a non-invertible matrix?

I'm trying to solve some systems of polynomials. I'm using Newton's method as described here. But my code breaks if my initial guess or any subsequent iteration lands on a point which is a solution ...
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1answer
48 views

How does the modified Newton's method for multiple roots converge quadratically?

I am currently learning about the multiple root issue for Newton's method. My textbook hinted at tweaking the function using $$h(x)=\frac{f(x)}{f′(x)}$$ and making the iteration function to actually ...
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1answer
47 views

newton's method on $\frac{1}{x}$

I was trying to use newton's method to approximate $\frac{1}{x}$, by finding the solution to $\frac{1}{x}=0$,I got $$x_{k+1}=x_k-\frac{\frac{1}{x_k}-n} {-\frac{1}{x_k^2}} = 2x_k-nx_k^2$$ I was trying ...
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2answers
84 views

Find all roots of a quintic polynomial given 1?

Let's say I have solved for one of the roots $a$ of a quintic equation using Newton's method. I read somewhere that you can "simply" divide the equation by $x - a$ to get a quartic and solve ...
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26 views

Computational complexity of $1/(x+z)$.

Is there a fast method to calculate $1/(x+z)$ where $z$ is a root of unity and $x$ is real. By fast computation, I mean is there a faster method than Newton-Rhaphson method.
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1answer
35 views

Newton Raphson for nonlinear overdetermined equations on noisy data [closed]

I would like to know, whether any improved Newton Raphson method is available for non-linear overdetermined equations (So we use Jacobian matrix and pseudo inverse). Data used as measurements are ...
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1answer
61 views

Newton's Method on $f(x)=\arctan(x)$ with oscillation

Consider the function $f(x)=\arctan(x)$ defined on $\mathbb{R}$. Let $x_0\neq0$ be an initial guess for the root of $f$ and apply Newton's Method with $x_0$. Further assume that the iterations produce ...
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29 views

Newton-Raphson converges with rate higher than 2? [duplicate]

Can we find an example of a function $f$ such that Newton-Raphson converges with rate higher than 2?
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23 views

How to prove that Newton's method for optimization will converge in one step

I'm studying Advanced motion control and we use Newton's method for iterative optimisation to find the minimum (optimal) control input. So my question is, Consider the following iterative update ...
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1answer
69 views

How to estimate distance from root to nearest immediate basin boundary for Newton's method in one complex variable?

Context: I want to check that the atom domain size estimate is smaller than the inradius of the Newton immediate basin, for centers of hyperbolic components in the Mandelbrot set, and thus justify ...
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1answer
57 views

Find the parameter $a$ and the coordinate $x$ of a function so that the function's maximum has a given value

I have implemented in C++ the code for a quasi-Newton method given by $x_{n+1} = x_n - \dfrac{\delta f(x_n)}{f(x_n+\delta) - f(x_n)}$ for a computational project in which $f$ is a function given by: $...
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3answers
61 views

Sequence of iteration in Newton Raphson method.

To find positive square root of $a>0$ by solving equation $x^2-a$ by Newton-Raphson mathod, if $x_n$ denotes $n$th iterate with $x_0>0, x_0\neq\sqrt{a}$, then the sequence $(x_n) $ is $1.$ ...
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1answer
34 views

How to find recurrence relation for $f(x)=x^3-\alpha=0$ where $\alpha>0$ using newton raphson method? [closed]

Alright for $x^3-\alpha=0$ we know that $x=\sqrt[3]\alpha$ is the root of course but unlike defined number it is too complicated to be solved using $x_0=\sqrt\alpha$ or $x_0=1$ for example so what is ...
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96 views

Round off error in modified Newton-Raphson

Newton's method for approximating roots of $f(x)=0$ is given by $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}, \quad n=0,1,2,\ldots,$$ where $x_n$ denotes the $n$th approximation of the root, and some ...
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36 views

Gradient descent vs Newton's method

rn im a little confused with gradient descent because it looks pretty similar to Newton's method but it does alpha*the derivative, instead of something like alpha/derivative which means that its ...
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1answer
33 views

Latent Dirichlet Allocation: Newton-Raphson method

I am trying to understand one step in the Newton-Raphson method, used in the paper on LDA by Blei, Ng and Jordan. Namely, how does taking the derivative of $L$, w.r.t. $\alpha_j$ result in this ...
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1answer
23 views

Thoughts on Bézier curve intersection and reliability of the test

Suppose we have two planar parametric curves f(t) and g(t), which are guaranteed to be second- or third-degree Bézier curves, which means they can be placed in "polynomial" form: $$ \textbf ...
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1answer
56 views

Find correct to 6 decimal places, the x-coordinate of the point on the curve y = ln x which is closest to the origin.

I must use the Newton Raphson method. Can someone please explain the steps I must take in order to reach the answer. For alot of this im having to self study the material to catch up and without ...
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25 views

Multivariate taylor to an expansion of Newton-Raphson.

Multivariate Taylor with 2 parts: $$\vec F(\vec x)=\vec F(\vec a)+\nabla\vec F(\vec a)\cdot(\vec x-\vec a)$$ Inverse Taylor which is the multivariate Newton-Raphson method: $$\vec x =\vec a + (\nabla \...
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3answers
92 views

Examples of when Newton's Method will fail?

I'm currently working on Newton's Method, and my instructor gave three instances where Newton's Method will fail. (A) Newton's method converges to another solutions x=b such that f(b)=0 instead of ...
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39 views

Minimizing the variance of arclength

For my computer science project I am working on a calculator that will find(or approximate) the point of equidistance along the surface of the Earth from N number of points. Using basic vector math I ...
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1answer
43 views

Newton's method and rearranging [closed]

I am new to root-finding and using numerical methods. Using Newton's method below: $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_0)}$$ using this chord formula where the chord length $c$ is 1 cm: $$c=2r\sin\frac{\...
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31 views

Sensitivity of starting point for Newton's method

I have been trying to solve the following minimization problem using Newton's method: $\min _{\boldsymbol{x} \in \mathbb{R}^{2}} f(\boldsymbol{x})=x_{1}^{4}-2 x_{2} x_{1}^{2}+x_{2}^{2}+x_{1}^{2}-2 x_{...
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41 views

Find the order of convergence of the Newton-Raphson method from a given table [duplicate]

I am given this table (Iterações = Iterations; Erro absoluto = Absolute Error) The question is: Determine the order of convergence of the method from the values of the table and justify. I got as ...
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25 views

Question on partial derivative derivation

I am a math beginner and I am trying to exercise on this Newton formula. I came across this question and was stuck in the steps. So please help, thank you in advance. For the following trend model, $Y=...
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1answer
34 views

Newton's Method for initial approximation

In order to find an approximation to the root of the equation $-x^3-\cos x=0$ by using Newton's method, which of the following initial approximations can be chosen? $x_0=1 , x_0=0, x_0=2 , x_0=1$ or $...
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59 views

Convergence of (Modified) Newton's Method with Roots of Multiplicity > 1 - Misunderstandings

I have been studying some numerical analysis and am currently looking at the order of convergence for Newton's method for finding roots with multiplicity more than 1; however, I have some ...
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52 views

Need help with Newton's method to find third approximation

Use Newton's method with the specified initial approximation $x_1$ to find $x_3$, the third approximation to the solution of the given equation $$x_5 = x_2 + 4,\quad x_1 = 1.$$ Find the value of $x_3$....
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2answers
79 views

Newton's method : Show inequality

I have written the formula of Newton's method to appoximate $a^{1/n}$, which is $$x_{k+1}=\left (1-\frac{1}{n}\right )x_k+\frac{a}{nx_k^{n-1}}$$ Now I want to show the inequality $x_{k+1}-a\leq \left (...
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41 views

Determinant Minimization Problem

I am trying to optimize the following problem $$\text{ min }\text{log}|\sum_{i=1}^{d}c_{ii}I^{(i)}|+s^{T}c$$ subject to $$\sum_{i=q+1}^{d}c_{ii}-\sum_{i=1}^{q}c_{ii}-t_2\ge 0$$ $$\sum_{i=q+1}^{d}c_{ii}...
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1answer
81 views

Improving convergence of the Newton-Raphson method

How can the Newton-Raphson method (that is, the multivariate generalization of Newton's method, used in the solution of nonlinear systems) be improved so as to attain better convergence? As-is, in ...
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27 views

Implementing Newton-Raphson method to find strike price in Black-Scholes but the error value keeps increasing?

I'm essentially writing VBA code to root solve for the strike price (K) given a premium (C or P). However, when I'm writing this code in VBA, I'm finding that the relative error term keeps on ...
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1answer
91 views

Number of iterations to find the root of $x^3+2x-54$ using Newton's Method

I am asked to calculate the (theoretical) minimum number of iterations needed to find the root $\alpha$ of $x^3+2x-54$ using Newton's Method, guaranteeing an absolute error less than $10^{-8}$, and ...
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3answers
35 views

How do I solve a nonlinear function with Newton-Raphson assuming that my variable is discrete?

Let's say I want to approximate $\sqrt 2$. I, therefore, write my function $$ f(x) = x^2-2 = 0 $$ I define $x$ to be a discrete variable which can take values $\color{brown}{0, 0.1, 0.2, 0.3, 0.4, 0.5}...
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2answers
38 views

Is Newton-Raphson the best we can do if we only know the derivative?

I was wondering if NR is the fastest method to find a root if all we know about a function is how to evaluate it and its derivative at any point. Since you can use the first derivative to approximate ...
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1answer
20 views

Newton tangent method (1D)

Newton tangent method for finding root of $f(x)=x^2-a \;$ is $$x_{k+1} = x_k-\dfrac{x_k^2-a}{2x_k},\;\;(k=0,1,2,...)$$ and its error $$|x_k-\sqrt{a}|\leq \dfrac{M}{2m}|x_k-x_{k-1}|^2$$ where $M=\...
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1answer
46 views

Mathematics iteration

Hi I recently was learning the newton raphson method but before I started learning that I came across a method for solving equations that cannot be solved using formulae or other methods called ...
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1answer
39 views

Newton-Raphson termination criteria for large problems

When using Newton-Raphson to solve a system of equations of the form: $$\mathbf{r}(\mathbf{x})=\mathbf{0}, \quad \mathbf{r}=[r_1, r_2,...,r_N], \quad \mathbf{x}=[x_1, x_2,...,x_N]$$ The termination ...
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1answer
48 views

Taylor series expansion about point $x_0 + \epsilon$

The taylor series expansion about point $a$ for function $f(x)$ is given by $$f(x) \approx f(a) + f'(a)(x-a) + \frac{1}{2!}f''(a)(x-a)^2 +...$$ As more and more terms are added to the taylor ...
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47 views

Higher dimension Newton Method

I am given this problem by my tutor in which the scope is to solve the linear system f1(x1,...,xd) = 0,....., fd(x1,...,xd)=0 and the first step is the following: 1.Fix x,y ∈R^d and let g(λ):=fi(x+λ(y−...

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