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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.

The Newton–Raphson method in one variable is implemented as follows:

The method starts with a function $f$ defined over the real numbers, the function's derivative $f'$, and an initial guess $x_0$ for a root of the function $f$. If the function satisfies the assumptions made in the derivation of the formula and the initial guess is close, then a better approximation $x_1$ is $$x_1 = x_0 -\frac{f(x_0)}{f'(x_0)}$$

The process is repeated as

$$x_{n+1} = x_n -\frac{f(x_n)}{f'_n(x_n)}$$

until a sufficiently accurate value is reached.

798 questions
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what are the failure cases of Newton Raphson Method?

I want list of failure cases for Newton-Raphson method. If possible please provide flow chart for Newton-Raphson method.
abcdmath
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Perform a graphical analysis of $f(x)$ to determine the starting values for which the iteration will converge.

Devise a Newton iteration formula for computing $\sqrt[3]{R}$ where $R>0$. Perform a graphical analysis of your function $f(x)$ to determine the starting values for which the iteration will converge. Answer: Let $R>0$ and set $x=\sqrt[3]{R}$. Then…
Username Unknown
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Newton-Raphson method with $c_0=3$ to calculate $c_1$.

I have a small problem Calculate a polynomial function $g$ with integer coefficient that has $\sqrt[3]{28}$ as a root, and then use the Newton-Raphson method with $c_0=3$ to calculate $c_1$. I now the Newton-Raphson…
Lain Symon
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Newton's method for $\bigl(\begin{smallmatrix}x^2-y^2+1\\ 2xy\end{smallmatrix}\bigr)=\bigl(\begin{smallmatrix}0\\0\end{smallmatrix}\bigr)$?

How do I find first four steps of Newton's method if $f(x,y)=\begin{pmatrix}x^2-y^2+1\\ 2xy\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}$ and $(x_0,y_0)=(1,1)$ ? Let $x^2-y^2+1$ be g(x) $\frac{\partial g}{\partial x}=2x$ and $\frac{\partial…
Alen
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Division-less computation of $1/(x+d)$ starting from $1/x$

I need to compute $\frac{1}{x+\Delta x}$ avoiding divisions, is this possible through an iterative method starting from $\frac{1}{x}$? I thought of two possibilities that could be different from the classical computation of the reciprocal using the…
KBowser
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What have I done wrong? in Newton's method calculation?

So I am using Newton's method. And for equation $f(x)=\sin(x)/(3+\sin(x))$ I differentiated it and I got $3\cos(x)/(3+\sin(x))^2$ which is checked by wolfram alpha. However, when I apply Newton's method on my own my results do not match my expected…
Tommy Lassa
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Newton's Method convergence to the closest $x$?

When applying Newton's algorithm, we need to pick an initial guess $x_0$ to start the process. My question is ... if there is convergence to a root, is this always the root that was closest to $x_0$?
Chach
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Using Newton-Raphson method to determine value at $t$ when $y=0.880$

A Fourier analysis of the instantaneous value of a waveform can be represented by: $$y=(t+\dfrac{π}{4})+\sin(t)+\dfrac{1}{8}\sin(3t)$$ Use Newton's method to determine the value of $t$ near to $0.04$, correct to 4 decimal places, when the…
Modrisco
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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 start with a horizontal tangent line (i.e when the…
MaximeJaccon
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Derive the error formula

I know that Newton's method is $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$ for n $\geq$ 0. Now, if I consider an interval [a, b] and I suppose that there exists a root $\alpha \in (a, b)(f{(\alpha)} = 0)$. I also assume that both f' and f'' are continuous…
Happiness Ia
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How to find initial guess for Newton-Raphson method

I have a polynomial equation $x^3-2x^2-5=0$ for which I have to find solutions accurate to $10^{-4}$ in the interval $[1,4]$. How can I find initial guess $P_0$?
the19thcenturydeveloper
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Difference between Newton's method and Modified Newton's method?

$X_{k+1}=X_k-H(X_k)^{-1}*G(X_k)$ This is what I know to be a newton Raphson method. However, one of my hw question is asking about Modified Newton's method, and what we have learned in class is Quasi Newton's method . Are Quasi & Modified same…
Young Seo Kim
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Is $f'>0$ enough for Newton's method to converge?

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 points?
joseMoralez1888
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Newton's method for finding square roots

Let us consider functions of the form $$ f(x) = x^{2} - a, $$ now the task is to find a root of f. I was asked to prove that for every initial guess $ x_{0} > 0 $ it is true that $ x_{1} \geq x_{2} \geq ... \geq \sqrt a $ and that the limit $…
Artur Wiadrowski
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Prove that Newtons Method works for a concrete polynomial

Can somebody help me with following: Let $f(x)=x^5-x+1$ and let $x$ be the limit of the Newton series $x_n=x_{n-1}-\frac{f(x_{n-1})}{f'(x_{n-1})}$. How do I prove that $x$ is a root of $f$, i.e. $f(x)=0$. I have tried to proove that…
Simon
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