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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.

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Convergence of Newton's method (exercice)

I am doing an exercice to prove the convergence of the Newton's Method. Let f b an holomorphic function on a simply connected space. $\ x^* \epsilon \space \Omega, \space f(x^*) = 0 \space $and $\ \space f'(x^*)\neq 0 $. $\ g = id -…
Hugh
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Numerical Solution: Minimise $d=\sqrt{x^2+y^2}$

I am trying to find the time of greatest solar eclipse for the forthcoming Eclipse on August 21st. I have 2 polynomials $$x = at^3+bt^2+ct+d \\ y = et^3+ft^2+gt+h$$ $a,b,c,d,e,f,g,h$ are constants published by NASA and $t$ is the time of the…
Nivag Swerdna
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How many iterations Newton's method needs

$g(x) = \frac{1}{2}(x - \frac{a}{x^2})$, $a > 1$ and $a<100$, if $a>0$ then $1<10^{2k}a<100$. So that $\sqrt{a}=\sqrt{10^{-2k}10^{2k}a} = 10^{-k}\sqrt{10^{2k}a}$. $x_0 = \min(a,10)$, $\forall a >0 $ $\exists k \in Z$ so that $10^{2k}a \in…
Marko Taht
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How to use Newton's method to solve a equation

I don't understand how to use Newton's method for this question. Consider the function $f(x) = 3 − x − e^x$ (i) Find the equations of the tangent and the normal to the curve $y = f(x)$ at the point $P(0, 2)$. (ii) Use Newton’s method in 2 steps to…
Maggie
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Approximation of convergance rate of Newton Method

I've just started calculus, so the phi function and such are perhaps a year away or such. I realize there are similar questions and perhaps duplicates. I'm don't know enough to do that. But here is my question. If only allowed pre- or very early…
Simon
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Newton-Raphson Method - Third positive root

I don't quite understand what is the difference between approximating the first positive root of a function and third positive root of a function by using the Newton-Raphson formula. Can the following formula apply to finding the third positive root…
phantom
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Why does newton's method of successive approximation to compute square roots work?

The algorithm is given as follows: Begin by guessing that the square root is x / 2. Call that guess g. The actual square root must lie between g and x/g. At each step in the successive approximation, generate a new guess by averaging g and…
Mutating Algorithm
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Newton's method error (with steps)

$f(x)=-x^3-2x^2-4x-25$ this is the function, and I found the derivative $f' (x)=-3x^2-4x-4$, and put it into Newton's method. I let the first term to be -4, and i got $x_(n+1)=-4+ 23/36 = -3.63....$ but from the website it sats the first term is…
Tommy Lassa
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Find the square root of $N = AB$.

Using Newton Raphson, show that root of the equation $N=AB$ is $$\sqrt {N}= \frac S4 + \frac NS, \text{ where } S=A+B.$$
ketankk
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Proof of the Newton Raphson method

I know only a little about this method. I get about the big picture. You draw a tangent from an appropriate point on function f, find the spot where it meets the x-axis, and draw another tangent from the point which has the same x coordinate on the…
Joshua Woo
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Understanding fixed point iteration

Claim that if at some interval $I$ show off $|\frac{f(x)f''(x)}{f'(x)^{2}}|\leq c < 1$ So then $x_0$ chosen from Interval $I$ results in an iteration of the fixed point of $g$. The base knowledge is that we know with Newton's method we can find the…
f1tz
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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(x+h)=f(x)+h f'(x)+\frac{1}{2} h^2 f''(x)) =…
user3431800
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Show that the equation $y=e^x/x^3$ has a root between $1.2$ and $1.3$

Show that the equation $y=e^x/x^3$ has a root between $1.2$ and $1.3$ Hey, having trouble with this one. Would appreciate a hand, not sure how to go about the exponent and such.
madbro13
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