Questions tagged [newton-method]

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

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Hessian-free preconditioner for non linear least squares

I am solving a nonlinear least squares problem using Gauss Newton method. Due to the large dimension of the problem, I use the Hessian-free approach. As a linear solver I use either MINRES or CG. To ...
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4 votes
2 answers
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Backward Euler + Quasi Newton(Broyden) method fails to solve Van der Pol's equation(Stiff ODE)

The first guess is using the forward Euler approach. The first jacobian is using finite differences. Then NR method is used to solve for the next iteration and Broyden's method is used to update the ...
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3 votes
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Applying displacement control loading using lagrange multipliers in the material non-linear finite element method

Hi I am trying to implement a simple plasticity based finite element code. I am not clear how to set up displacement control applied through Lagrange multipliers. In case of a linear problem, I did ...
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Viable algorithms for efficiently solving a block matrix system with non-uniform sparsity structure

I am trying to use Newton's method to get a stationary solution for a system of equations of the following form: $$ \begin{Bmatrix} \frac{\partial x}{\partial t} \\ 0 \end{Bmatrix} = \begin{Bmatrix} f(...
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6 votes
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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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Blown-up iterates in Gauss-Newton method

I am working on a non-linear least squares problem with standard form, in which I need to calibrate a parameter vector $\Theta$ to a set of inputs $\mathbf{x}$ and outputs $\mathbf{y}$: $$\begin{align}...
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Newton-Raphson for estimating Weibull distribution does not converge

I've been trying to estimate the two-parameter (a,b) Weibull distribution (loc. param. = 0). $$f(t;a,b)=\frac{b}{a}\left(\frac{t}{a}\right)^{b-1}\exp(- \left(\frac{t}{a}\right)^b) $$ To find the ...
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4 votes
3 answers
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Which absolute and/or relative stopping criteria do use for Newton's method?

I saw many stopping criteria for Newton's method all around Web and books. Some are defined from the residuals: of either current iteration only: $$ \|f(\mathbf{x}^{(k)})\| \leq \epsilon $$ (https://...
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1 answer
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Drawing saddle node bifurcation diagram for a non-linear ODE in Python

I'm trying to draw the bifurcation diagram of the following ODE, This ODE leads to a saddle-node bifurcation (see wiki) However what I get is not exactly right. There's a lot of "noise" as ...
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2 answers
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Numerical Methods of solving a non-linear ODE?

I want to solve the nonlinear equation $\frac{d^2x}{dt^2} + k\sin x = 0$, numerically. I found that solving this elliptic integral would be cumbersome, so is there a numerical method i could use to ...
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Solution predictors for accelerating convergence in nonlinear FEM

I am looking for the details of commonly-used predictors for accelerating the convergence of iterations using Newton-Raphson scheme for nonlinear problems in FEM. I am looking specifically for static ...
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Coding up Newton's method for a mapping from R^2 to R -- the Jacobian wouldn't be invertible

I'm trying to code up in Matlab a multivariable Newton's method, for a mapping from R^2 to R, but the Jacobian would be a 2x1 matrix, not square, so it wouldn't be invertible. Does this mean that ...
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2 votes
1 answer
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Solving system of nonlinear vector functions

I am trying to figure out how to implement a solver for a system of nonlinear equations of the form \begin{align*} u_1 &= y_n + h\left(a_{1,1}f(t_n + c_1 h, u_1) + a_{1,2}f(t_n + c_2 h, u_2)\...
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Quadratic optimization with nonlinear vector term

I wish to minimize the quantity $$W=1/2x^TAx-x^Tg(y)$$ with respect to $x$ and $y$, which are vectors of unknowns. $A$ is a sparse square symmetric positive definite matrix and $g(y)$ is a vector with ...
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Convergence of Truncated Newton for non-convex Hessian

I was wondering if anyone could enlighten me about the convergence properties of the truncated newton method in case of a non-positive definite hessian $\nabla^2 f = H$. From the Book 'Numerical ...
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How to check curvature of a vector valued function

In terms of numerical optimization, the newton-rapson method requires a pos. definite Hessian $\nabla^2f$ respectively pos. curvature for computing the next step $p_k$ by solving $$\nabla^2 f p_k = -\...
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2 votes
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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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3 votes
1 answer
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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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1 answer
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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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2 votes
1 answer
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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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1 answer
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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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243 views

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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1 answer
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Solving nonlinear PDE with finite difference based on Newton-Krylov

I am now working on solving MHD equations with finite difference method, which include nonlinear equations: $$ \frac{\partial\rho}{\partial t}+\nabla\cdot\left[\left(\rho_0+\rho\right){v}\right]-\...
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2 votes
1 answer
531 views

GMRES vs Newton-GMRES for Solving nonlinear PDE's

Often when numerically solving nonlinear PDE's using method of lines approach with an implicit integrator a system of nonlinear equations have to be solved. To be more specific, let's say we have ...
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Residual value goes to NaN while solving a system of nonlinear equations

I am solving a system of coupled nonlinear equations using Newton's method, similar to $$\begin{split} c_A(A, B)\partial_tA&=\nabla\left(k_A(A, B)\nabla A\right) + f_A(A, B, t)\\ c_B(A, B)\...
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Positive root of $x^q + bx - b$

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\...
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1 vote
1 answer
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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}\\...
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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-...
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1 answer
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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-...
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2 votes
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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{...
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2 answers
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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)...
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6 votes
1 answer
620 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 ...
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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 ...
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3 votes
1 answer
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Computational complexity of Newton's method

the classical Newton's method for non-linear systems of equations is $x_{k+1} =x_k-J_F(x_n)^{-1} F(x_n)$. In pratice, rather than compute the inverse of the Jacobian matrix, one solves the systems $...
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9 votes
0 answers
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Imbalance of variables in Mixing Newton's method and Linear solver for a Non-linear system

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{...
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1 vote
1 answer
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Newton's method problem

I struggle with this problem: I understand how to apply Newton's method to a already given function but I can't seem to find that function in this case from the given input n, t, d and s. I'm ...
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1 answer
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Root finding using Newton's method with quadratic interpolation - Is there a mistake in this textbook?

I've been trying to learn about root finding using Newton's method, which uses a quadratic interpolating polynomial. I found this text, Numerical methods for roots of polynomials, PART 2: McNamee &...
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1 vote
1 answer
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Muller's method is the same as Newton's method with a quadratic interpolating polynomial?

I'm new to numerical analysis, and have been learning root finding algorithms. I am a bit confused about the difference between Muller's method, and Newton's method using an n-degree interpolating ...
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3 votes
0 answers
315 views

Difference between Dishonest Newton method and Very Dishonest Newton method

What is the difference between the Dishonest Newton method and the Very Dishonest Newton method? Is there a difference or do they mean the same thing? I have tried searching for this on the internet ...
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2 answers
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Which SciPy nonlinear solver when Jacobian is analytically known and sparse?

I have a nonlinear function fun with n inputs and n outputs. I also have a function jac which calculates the Jacobian, which is ...
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2 votes
0 answers
429 views

How to prevent BFGS from getting stuck on astronomically large gradient?

I have implemented BFGS myself from scratch in order to solve minimization problems. Part of BFGS, as I understand it, is that the approximation to the Hessian is supposed to be positive definite, ...
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3 votes
2 answers
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How does one calculate reaction force in FEA?

I wrote a UEL (User Element in Abaqus) for one element and compared to a reference UEL which used standard FEM, where the results agreed satisfactorily, except the reaction force. The stress, strain, ...
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5 votes
1 answer
144 views

Can redundant variables be beneficial for root-finding convergence

Suppose I have $n$ generally nonlinear equations for $n$ variables, like e.g. for $n=2$ the system $F(x,y)=0$ $$ \begin{aligned} x^2+2y-4&=0\\ \sqrt{8}x+y^2-5&=0 \end{aligned} $$ By ...
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Nédélec Elements and Newton-Methods

If you want to develop numerical algorithms for variational inequalities, you often choose a Semismooth Newton Method. In many cases, this approach involves derivatives of $\max$ or $\min$ functions ...
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7 votes
2 answers
554 views

How to calculate/derive analytic FEM Newton Jacobian

I trying to wrap my head of derivation of the analytic FEM Jacobian for the Newton method. Say we have a nonlinear Poisson problem of the (weak) form $$ \int a(u)\nabla\ u\cdot \nabla v = \int f v $$ ...
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Vectorised root finding in Python

I have an array of size (254, 80) which I am trying to use Scipy's fsolve on. I have found that the speed of using fsolve on a vector is quicker than it is in a for loop but only for vectors upto ...
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