Mathematics

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.

844 questions
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Why is it not possible to generate an explicit formula for Newton's method?

Going through the recursive formula for approximating roots every time is extraordinarily tedious, so I was wondering why there was no formula that computed the $n$th iteration of Newton's method.
user2707299
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Bounding the basins of attraction of Newton's method

In general, Newton's method for root finding has a "bubbly" boundary between basins of convergence for different roots. This is where fractals are usually created from. But outside these "bubbly" boundaries there are very clear areas where there's…
Jay Lemmon
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Pattern of Newton-Raphson iteration $x\mapsto\frac{1}{2}(x+\frac{q}{x})$ over finite fields

While playing with Newton-Raphson method over finite field $\mathbb{F}_p$ , I noticed some cute patterns that I can't explain out of my brain contaminated with analysis. Here is the setting: Setting. Let $p$ be an odd prime and $q \in \mathbb{F}_p$…
Sangchul Lee
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Convergence of the quadratic map $\left(x-\left(x-\left(x- \dots \right)^2 \right)^2 \right)^2$?

Edit - I changed the title and much of the body to better reflect my full question. The old one I don't really care about, although I appreciate Fabian's answer of course. Here is the plot for the function (or map), defined as follows:…
Yuriy S
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Newton's method intuition

In optimisation the Newton step is $-\nabla^2f(x)^{-1}\nabla f(x)$. Could someone offer an intuitive explanation of why the Newton direction is a good search direction? For example I can think of steepest gradient decent as a ball rolling down a…
Daniel
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Newton's method for square roots 'jumps' through the continued fraction convergents

I know that Newton's method approximately doubles the number of the correct digits on each step, but I noticed that it also doubles the number of terms in the continued fraction, at least for square roots. Explanation. If we start Newton's…
Yuriy S
  • 32,998
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Convergence of Newton Method for monotonic polynomials

Consider a polynomial $p : \mathbb R \to \mathbb R$ with $p'(x) > 0$ for all $x \in \mathbb R$ . The function $p$ has exactly one real zero. Will the Newton method $$x_{n+1} = x_n - \frac{p(x_n)}{p'(x_n)}$$ converge for all $x_0 \in \mathbb R$ ?…
flawr
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does this Newton-like iterative root finding method based on the hyperbolic tangent function have a name?

I've recently discovered that modifying the standard Newton-Raphson iteration by "squashing" $\frac{f (t)}{\dot{f} (t)}$ with the hyperbolic tangent function so that the iteration function is $$N_f (t) = t - \tanh \left( \frac{f (t)}{\dot{f} (t)}…
crow
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When does Newton-Raphson Converge/Diverge?

Is there an analytical way to know an interval where all points when used in Newton-Raphson will converge/diverge? I am aware that Newton-Raphson is a special case of fixed point iteration, where: $$ g(x) = x - \frac{f(x)}{f'(x)} $$ Also I've read…
Anonymous Gal
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Solving $\ln{x}=\tan{x}$ with infinitely many solutions

Lets take $f(x)=\ln{x}$ and $g(x)=\tan{x}$ When $f(x)=g(x)$ that is $\ln{x}=\tan{x}$, we see that the graph is like: Hence we see that there are infinitely many solutions to $x$ but the two graphs do not coincide (like while solving "$x=x$"!) The…
NeilRoy
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How to solve $x^2 = e^x$

The question is to find $x$ in: \begin{equation*} x^2=e^x \end{equation*} I know Newton's method and hence could find the approx as $x\approx -0.7034674225$ from \begin{equation*} x_{n+1} = x_n - \dfrac{x_n^2-e^{x_n}}{2x_n-e^{x_n}} \end{equation*}…
NeilRoy
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Reference request for Newton's method on a convex/concave function

The proof featured at http://planetmath.org/newtonsmethodworksforconvexrealfunctions shows that, roughly speaking, Newton's method is unconditionally convergent for convex/concave functions . I am looking for a textbook or paper reference of this…
parsiad
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Understanding the connection between higher-order Newton methods and the convex-concave procedure

I've been studying a paper Ahmadi et al . $^\color{magenta}{\dagger}$ that introduces higher-degree Newton methods with improved oracle complexity compared to the classical approach. This paper seems to be very interesting and has recently appeared…
Tuong Nguyen Minh
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Why does multiplying by the inverse of the Jacobian matrix generalize the Newton-Raphson method to n dimensions?

I don't fully understand how multiplying by the inverse of the Jacobian matrix generalizes the Newton-Raphson method to $n$ dimensions. In other words, I don't understand how this: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ is equivalent to this in 2…
Autechre
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Prove or disprove - Newton's method convergence in higher dimensions

It's not an exercise for uni or anything like that, just something that's been bothering me a bit and I can't seem to find useful information on the web on the matter. When talking about real valued scalar functions, we know that newton's method…
Oria Gruber
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