# 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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### 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 ...
1 vote
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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: 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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### 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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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 ...
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) ...