# Questions tagged [newton-method]

Method for finding successively better approximations to the roots (or zeroes) of a real-valued function

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### continuous analogues of Newton's method

Suppose we want to minimize some convex functional $J(u)$ where $u$ lives in some Banach space $V$. The classical Newton method $$\mathrm d^2J(u_n)(u_{n + 1} - u_n) = -\mathrm dJ(u_n)$$ can be viewed ...
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### tiny typo in Numerical Recipes Eq. 9.4.6 [closed]

The Numerical Recipes Forum http://numerical.recipes/forum/ is closed, so I will record a tiny typo here for the benefit of others who may wonder about this. (This typo is not in the software, but ...
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### Constrained Newton-Raphson root finding

Original Question I have a set of non-linear equations and I need to find the root where a subset of my solution vector is constrained to be greater than or equal to 0. I have implemented the Newton-...
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### Inverse Newton Method for optimization: is this the correct algorithm?

I am trying to implement the algorithm in this article. I have already asked a question before about it here, and I am trying to figure out what I am doing wrong. This time, it's this section of the ...
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### How do I globally change the precision of a piece of code in Python to debug it?

I am solving a system of non-linear equations using the Newton-Raphson method in Python. This involves using the solve(Ax,b) function (...
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### Parallelizing Newton-method in solving non-linear systems

Circuit simulation software based on SPICE (such as ngspice) uses Newton-Raphson method to solve non-linear system of equations ...
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### How to avoid gsl root finder evaluate function outside its domain

When I use the newton's method or hybrid solver in the GSL package to deal with 1-D or multidimensional root solving problems, the code frequently crashes when the solver requests function value ...
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### Evaluation of slope at iteration ith - Newton-Raphson method

I'd like to know how Ansys computes the slope (=stiffness matrix) at point x1 in figure. I'm studying the way in which Ansys uses the Newton-Raphson method when there are nonlinearities. In the slide ...
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### Number of GMRES iterations increase when stepping forward in time, using the Newton method

I am solving a system of nonlinear time-dependent equations using the Newton method in a finite-element-setting, i.e. first I create the jacobian matrix for the current time, and afterwards I try to ...
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### How to use Newton-Raphson method to handle nonlinear terms in coupled system of PDEs?

I'm trying to solve the Nonlinear Schrodinger's Equation (NLSE) in 2D using Finite Elements, but I don't know how to handle the nonlinear term. I suppose I have to apply the Newton-Raphson algortihm ...
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### How to use Wolfe-Powell step-size control in quasi-Newton method?

I'm trying to find the minimum of a function using the quasi-Newton method with the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. But I want to change the following implementation, so that: 1) ...
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Is there either a closed-form expression or fast/elegant algorithm for computing the positive root of the polynomial $$f(x)=x^q + \beta x - \beta,$$ where $\beta>0$ and $q\geq2$? How about the $q\... 1 vote 1 answer 113 views ### Product of rank one updates as a low rank update for quasi newton/BFGS I'm trying to improve the speed of the following iteration to calculate$s_k$: $$B_k^{-1} = \Bigg( I + \frac{s_{k}s_{k-1}^T}{||s_{k-1}||^2}\Bigg)...\Bigg(I+ \frac{s_1s_0^T}{||s_0||^2}\Bigg) B_0^{-1}\\... 0 votes 1 answer 119 views ### Debugging Newton-method used in a CG-approach I am currently proof-checking my program, which is intended to use Newton's method for solving nonlinear equations, using a continuous galerkin approach. Thus, as first step I checked it using a time-... 7 votes 1 answer 604 views ### Lack of quadratic convergence in Newton's method It is well-known that Newton's method can converge quadratically, if initial guess is close enough and if the arising linear systems are solved accurately. I am applying Newton's method to highly ill-... 2 votes 0 answers 115 views ### Nonlinear system with diagonal nonlinearity Consider a nonlinear system of the form \boldsymbol{f}(\boldsymbol{x}) = \boldsymbol{0}_{\mathbb{R}^n} for \boldsymbol{x} \in \mathbb{R}^n, where the function \boldsymbol{f} is given by \begin{... 1 vote 2 answers 663 views ### Implementation of the Jacobian-free Newton method In my calculation (of a simple heat equation, for testing) using the Newton method, I tried to replace the full Jacobian matrix with an approximation vector, i.e. replacing J in$$J(u)\delta u=-F(u)... 6 votes 1 answer 886 views ### Defining a condition number and termination criteria for Newton's method The condition number of function evaluation $$\mathrm{cond}(f,x) := \left| \frac{x f'(x)}{f(x)} \right|$$ is infinite at a root of$f$. Hence it is useless for rescaling a tolerance which defines an ... 1 vote 1 answer 311 views ### Shooting method implementation I have a trouble of defining a Jacobian matrix for my problem. Basically, I have 4 differential equations to be solved.$\begin{aligned} \dot x_1(t)&=x_2(t)\\ \dot x_2(t)&=p_2(t)−\sqrt 2 ... 3 votes 1 answer 2k views ### Computational complexity of Newton's method the classical Newton's method for non-linear systems of equations isx_{k+1} =x_k-J_F(x_n)^{-1} F(x_n)$. In pratice, rather than compute the inverse of the Jacobian matrix, one solves the systems$...
Problem Solving a non-linear system of equations. The number of variables is the same as the number of equations. When I fix a set of variables (say $\vec{y}$) and keep another set free (say \$\vec{...