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 ...
- 3,095
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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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How to iterate $(x-1)/(x\ln{x})=a$ using Newton Raphson
I am trying to iterate $\frac{x-1}{x\ln{x}} - a= 0$ to solve for $x$ using Newton Raphson, but it is blowing up. Any ideas why? Is this a strange function to find a root for?
$$x_{i+1} = x_i - f(x_i)/...
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Calculating convergence order for Newtons method
I am trying to teach myself numerical analysis and I am trying to find out how you determine the order of convergence for newtons method. On the wiki there are multiple formulas for newton methods ...
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4
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How to determine the almost accurate initial guess to calculate the $\ln(x)$ via Newton-Raphson method?
I'm developing my own Real Number class, and for that, I have to implement the Natural Logarithm function.
I've used Maclaurin series of Natural log with iteration of 300 or 500. The division method ...
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Order of convergence/convergence rate of variant of Newton's Method [duplicate]
$x_{n+1}=x_n - \frac{f(x_n)}{f'(x_n)}-\frac{1}{2}\frac{f(x_n)^2 f''(x_n)}{f'(x_n)^3}$
For obtaining the formula above, I did the following:
$f(x_n+ \Delta x_n) \approx f(x_n) + f'(x_n)\Delta x_n + \...
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Is there a self-correcting iterative method for approximating pi without using transcendental functions?
The Newton-Raphson method is an iterative method for finding a root of a function, and it is self-correcting in the sense that any error in the initial input is reduced with each iteration so that it ...
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Aitken's $\Delta^2$ Method applied to the Newton's Method
Could Aitken's $\Delta^2$ Method be used to accelerate the convergence of the Newton's Method? I simplified $p_n$=$x_n$-$\frac{(x_{n+1}-x_n)^2}{x_{n+2}-2x_{n+1}+x_n}$ and obtained $p_n$=$x_n-$$\frac{(\...
- 333
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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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Newton method with high multiplicity: rigorous proof
I am looking for a rigorous yet reasonably short proof of the following statement: let $f\in \mathcal{C}^{m}([a,b])$, and let $\alpha \in (a,b)$ such that
$$
0 = f(\alpha) = f'(\alpha) = f''(\alpha) = ...
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Vector equation where each element is equal to a quadratic form
Is there a way to solve an equation of the following form:
$$x_i = {\bf x}^T{\bf A}_i{\bf x} + b_i$$
where $x_i$ is the ith element of vector $\bf x$.
I'm guessing this could be equivalently expressed ...
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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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Is there a Newton-Raphson method for finding the zeros of scalar functions of several variables?
I have seen countless examples of using the Newton-Raphson method to find the roots of a system of equations. But I cannot find anything on the possibility of finding the zeros of a scalar function of ...
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Discontinuities in a system of equations solve - Hydraulic ram end stops
I am trying to implement hydraulic ram end stops/limit stops, this being one component within a wider hydraulic model. Each component contributes residual equations such as force equilibrium, mass ...
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When does higher-dimensional Newton-Raphson converge?
If we know objective function $f:\mathbb{R}\to\mathbb{R}$ is concave-up, decreasing, and has a solution $x^*$ on interval $I$, (or equivalently, $f$ is concave-down, increasing, and has a solution $x^{...
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Finding all real roots of a polynomial by using Newton-Raphson method.
Is there a general formulation for finding all real roots of a polynomial by using the Newton-Raphson Method?
I have seen the answer from https://math.stackexchange.com/a/998489/1205840. This method ...
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Write Newton-Raphson for Lagrangian Navier-Stokes
I am trying to write the finite element formulation of the Cauchy momentum equation (Navier Stokes) in Lagrangian formalism:
$\begin{aligned}
&\nabla\cdot\sigma+\rho\mathbf{g}=\rho\dfrac{\partial\...
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Newton's method: partial or total derivative for composed functions?
Suppose I have the following Jacobian
$J_{ij}(\underline{\Delta\sigma})=\displaystyle\frac{\partial \Delta\varepsilon_i}{\partial \Delta\sigma_j} \in \mathbb{R}^{6\times6}$
within a Newton-Raphson ...
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Approximations for $n$th roots
Recently, I have asked how to approximate cube roots to at least one decimal digit. I want to do the same for $n$th roots ($n$ is an integer).
Here's what I have done:
Let $\sqrt[n]{a}$ be the number ...
- 5,032
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Convergence of Newton-Raphson given a generic polynomial
Let $f(x) = (x-r_1)(x-r_2)...(x-r_d)$ where $r_1 < r_2 < ... < r_d$. I need to prove that if $x_0 > r_d$, Newton-Raphson converges to $r_d$.
I could see that:
$$f(x) = \prod_{j=1}^d (x-r_j)...
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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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How to make nearly perfect assumption of $ x_0 $ to proceed further with Newton-Raphson method?
I'm writing my own RealNumber class in Python, where I need to get the $ n $-th root of $ M $. Here, $ n, M \in \mathbb{Z}^+. $
I'm trying to follow the Newton-Raphson method for calculation of the $ ...
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Searching Newton-Raphson failures!!
I came up with a numerical method to find roots for nonlinear functions. Now I'm trying to see how better from other methods is. So.... where does Newton-Raphson fail? I came up with a function $f(x)=...
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Newton Method for Scalar Field
I understand how Newton Method comes naturally from linear Taylor approximation of the function.
For $f: \mathbb{R}\to \mathbb{R}$ we have
$$
x_{new}=x+\Delta x
\\
f(x+\Delta x)=f(x)+f'(x)\Delta x=0
\\...
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Symmetric Trilinear map for proving Newton's method on self-concordance functions
The following claim is used for proving the quadratic convergence phase of Newton's method on self-concordance functions. I have a very long and non-intuitive proof using Lagrange Multipliers. I ...
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Non-pathological example where Newton's method converges to a saddle point or local maximum
It is well known that Newton's method (with full step size) may converge to a saddle point or local maximum for some initial values (assuming it converges to something), but I have trouble visualizing ...
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How to prove that the Newton-Raphson method is convergent in this problem? [closed]
Use the Newton-Raphson method to find the roots of the function
$f(x) = x^5 - 14.3x^4 + 76.15x^3 - 185.525x^2 + 202.3x - 89.625$, with an initial guess of $x_0 = 1$.
How can we prove the convergence ...
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Introduction to algorithms (CLRS) Problem 33-1 Newton's method
Here is the original problem
Gradient descent iteratively moves closer to a desired value (the minimum) of a function. Another algorithm in this spirit is known as Newton’s method, which is an ...
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Newton Method Error Bound
I am trying the following exercise:
Prove that the function $ f(x) = x^3 - 2x - 5 $ has a root, $ r $, at $ [2, \ 2.2] $. If we consider the succesion $ (x_n)_n $ defined by Newton Method starting ...
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Why does Newton-Raphson method oscillate between two values [duplicate]
If I use starting value x0 = 1 for the following function Newton-Raphson oscillates between -0.113356775 and 0.113356775. If I however use other values, I manage to find the root.
Why is that? Any ...
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roots of functions with cusps (combined waves)
If I have some non-negative functions with shapes similar to the one in the picture, (imagine, for simplicity, that each function is the sum of two $sin$ waves taken in absolute value.)
In order to ...
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Why doesn't Newton-Raphson work for these type of functions?
Suppose that for $f(x)$, $\frac{f(x)}{f'(x)}=3x$. If I wanted to find a root for $f$ using the Newton-Raphson method I have the formula $x_{n+1} = x_{n} - \frac{f(x_{n})}{f'(x_{n})} = x_{n} - 3x_{n} = ...
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Newton-Raphson Method with Sum Constraint
I have a system of non-linear equations $\pmb{f}\left(\pmb{x}\right)$ for which I want to find the approximate root $\pmb{x}^*$ using the Newton-Raphson method, $$\pmb{f}\left(\pmb{x}^*\right) \approx ...
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Newton's method for $\sqrt[n]{c}$
I'm studying the book of Elon Lages Lima, Análise Real volume 1. In the section of applications of derivatives, the author wants to estimate the error for calculating the nth root of $c>0$. Let $f(...
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Oscillations in Newton's fractal
I'm working on a program that draws Newton's fractal for a given polynomial. Newton's fractal is a fractal derived from Newton's root-finding method, which given some initial guess $x_0$ and function $...
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Why can't Newton's method be used directly for square roots?
If I want to use Newton's method for a polynomial function I simply use the definition and derive it to get to the fraction which needs to be deducted from the current guess. For instance, if $f(x)= x^...
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closed-form Newton flow of tanh(ln(1+x^2))
The differential equation for the Newton flow $z (t)$ of $f (t)$ is given by
\begin{equation}
\dot{z} (t) = - \frac{f (z (t))}{\frac{d}{d t} f (z (t))} = -
\frac{f (z (t))}{\dot{f} (z (t))}
\end{...
user1057203
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Proof of Newton method in higher dimension
I am struggling with the following problem related to Newton's method in higher dimensions. Hope somebody can give me some advice using the contraction theorem.
Let $f:\mathbb{R}^N \to \mathbb{R}$ be ...
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Is the Lambert W function the Newton flow of the exponential function?
Is this right?
The Lambert W function, denoted by $W(z)$, is defined as the inverse function of $f(z) = ze^z$. In other words, if $w = W(z)$, then we have $z = w e^w$.
The continuous Newton's method ...
user1057203
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Conditions for Convergence of Newton's method
My professor said that if we have a twice continuously differentiable real function $f$ in an interval $[a,b]$ such that:
$f(a)f(b)<0$
$f'$ and $f''$ don't change signs and $f'$ does not vanish
...
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What's the connection between the inverse function theorem and Newton's method?
Let $f: \mathbb{R} \to \mathbb{R}$ and consider the problem of solving $f(x) = y$.
The inverse function theorem says
$$ \mathrm{d} x= \frac{\mathrm{d} y}{f'(x)} $$
We could turn this ODE into a finite ...
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Newton Method with backtracking line search
I have a school exercise where I am supposed to implement the Newton method with and without a backtracking line search. The first one converges quite rapidly but, when I add the backtracking line ...
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Error Newton's Method
My profesor said that (if $\alpha$ is the root of the function) then $$|x_{k+1}-\alpha|\leq \displaystyle\frac{M_2}{2m_1}|x_{k+1}-x_{k}|,$$ where $M_2$ is the maximum of the second derivative and $m_1$...
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Newton's Method linear and quadratic convergence
I am learning Newthon's Method to Nummericaly find roots.
The book (Sauer, Numerical Methods) proves that it has a quadratic convergence for simple roots by rearranging the Taylor series for the ...
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