# 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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### A proof that Newton method converges

I am asked to write down the Taylor series for a function $f$ evaluated at $x + h$ in terms of $f(x)$ and its derivatives evaluated at $x$. Then, to use this result to show that if $x_0$ is an ...
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### Solving 5 equations with 5 variable using Newton Raphson method

I have 5 equations with 5 variables $X_1$, $X_2$, $X_3$, $X_4$, and $X_5$, namely \begin{align} a_{11}X_1 + a_{12}X_2 + a_{13}X_3 + a_{14}X_4 \sin X_5 &= b_1,\\ a_{21}X_1 + a_{22}X_2 + a_{23}X_3 + ...
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### Finding Newton method order of convergence

I'm trying to determine how you find the order of convergence of newton's method. I have the formula $$\frac {|x^*-x_{n+1}|}{ |x^*-x_n|^q} = \alpha$$ I'm setting $q=2$ to test for quadratic ...
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### Newton's Method convergence for all approximations in $[a,b]$

Given a function $F(x)$ defined on $[a,b]$ such that: $f$ $\in$ $C^2([a,b])$ $F(a)F(b)$<0 $F'(x) \neq 0$, $\forall$ $x$ $\in$ $[a,b]$ $F''(x)$ $\geq$ $0$ or $F''(x)$ $\leq$ $0$, $\forall$ $x$ $\in$...
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### Why does taking the tangent line improve the approximation in Newton's method?

I have gained a comprehension of the operational process through the discussion located at Why does Newton's method work?. Nevertheless, there is one aspect that remains unclear to me. To initiate,...
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1 vote
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### Newton's method: convergence

Suppose $f$ is differentiable in the interval $[a,b]$ and $x_0 \in [a,b]$. Given the Newton's method sequence $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$, with $x_n \in [a,b] \forall n$,show that if the ...
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### New depth of water in a cylindrical tank

A cylindrical tank has a diameter of $60 cm$ and a height of $100 cm$. It is filled with water to a depth of $10 cm$, then tilted by an angle $\theta = 30^\circ$. What will be the new depth of ...
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1 vote
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