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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.

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### Proof of Newton-Kantorovich theorem, Wikipedia version

Context I have recently been researching the Newton-Kantorovich theorem after wondering about convergence criteria for the Newton-Raphson method in numerical analysis, as it seems to be the most ...
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### How does extra equations affect Newton-Raphson method's performance on solving system of non-linear equations? [closed]

I'm working with model updating, in which the model's parameters are adjusted in order to reduce its ouput error in relation to a reference. For this, I would like to compare minimizing a single ...
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### Newton-Raphson Method's Convergence

I have a function with three real roots, for which I have to prove the following: There are three intervals in which, for every initial guess, N-R converges to the root. I have this Theorem from a ...
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### How can I derive equation 2.23(Newton-Raphson for Entropy) in NASA CEA analysis

It might sound a bit basic, but, I'd like to follow NASA CEA report I. analysis from the beginning. So, I have to derive the Newton-Raphson equation from the entropy equation, which is one of the ...
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### Tangent definition with a newton raphson question example

I'm very confused about the definition of a tangent, I was told its a straight line that touches a curve at a point, but if extended does not cross the curve at any other point. In this question if u ...
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### Sensitivity of Newton's method initial vlaue

Can someone explain to me what sensitivty refers to when it comes to root finding? When we say it is sensitive to initial value for x, is because there are multiple roots, or because there might be a ...
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### Newton's convergence for systems of equation analysis

So I solved a question using Newton's Method for systems of equations. Then they asked: How can you ensure that Newton's method converges as it should? What convergence rate do you observe? My idea ...
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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 ...
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### Newton's method - does it always work for functions that are close to the identity function? [closed]

Since Newton's method for finding roots trivially works for the identity function $t \mapsto t$, I was wondering whether it also works for every continuously differentiable functions that is ...
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### For a differentiable function $f$ , does this mean value theorem iteration method converge to the root of $f?$

Suppose $a<b,\quad f:(a,b)\to\mathbb{R}$ is a differentiable, nowhere-linear function$,\ f(c)=0$ for some $c\in (a,b),\quad$ and $f'(x)\neq 0\ \forall\ x\in (a,b).\$ Let $x_0=a;\ x_1=b.$ Nowhere-...
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### Infinite series expansion for roots

Newton's method states that, if there is a root $r$ of $f(x)$ that we want to calculate and $r_0$ is an approximation, then a better approximation is $$r_1=r_0-\frac{f(r_0)}{f'(r_0)}$$We could ...
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### Newton's method when $f'(\overline{X}) = 0$

Everywhere I read a proof of Newton's method (that one with $x_{n+1} = x_n-f(x)/f'(x)$), it is used that the derivative of the function does not vanish at the root we are looking for. There is a post ...
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### Simple modification of Newton's method problem

Since you can transform Newton's recurrence relation to the original equation $f(x)=0$ by considering the limiting case as $n$ tends to infinity, it's not surprising you can also do it the other way ...
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### variant of Levenberg-Marquardt suitable for Lagrange-Newton method

If a Newton descent is applied to some scalar function of a vector, the Hessian is positive definite in the vicinity of a minimum, but can become indefinite in larger distance from the minimum. This ...
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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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### Excel solver in newton-raphson

I have an excel worksheet that solves system of PDEs using newton-raphson method. The solution obviously depends on some variables that I input. There is a VBA macro (lets call it VBA1) that is used ...
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### How to find rational power of two using Newton's method?

I know how to find integer powers of two, $2^x = \prod_{i=1}^x 2$, I have memorized powers of two up to 32nd power of two, and I use bit-shifts to calculate them. For integer powers of other numbers I ...
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### How to find log2 of a number with Newton's method?

I am learning C++ and I find pow, log, log2, exp ...
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### Algorithm for non-linear system of equations

I would like some tips in figuring out a good algorithm to find the solution of the following system. Let $\theta$ be a constant in $(0,1)$, let $i,l=1,...,N$, let $a_{l}$ and $b_{i,l}$ be some ...
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