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Questions tagged [newton-method]

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

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1 vote
1 answer
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How to impose boundary conditions when solving a nonlinear dynamical system given by the FEM solver

I am solving a nonlinear dynamical system given by a nonlinear elastic problem which takes the following form: $$ \boldsymbol{M} \ddot{u} + \boldsymbol{K}_{\textrm{NL}}u = 0 ,$$ here $u \in \mathbb{R}...
Saddam N Y Hijazi's user avatar
1 vote
1 answer
78 views

Multivariable Newton's method for-loop

I am struggling with an assignment concerning Newton's method. We are to approximate 3 intersection points using intersecting circle coordinates P1, P2 and P3, using Newton's method. And I can do that ...
rawestan's user avatar
2 votes
0 answers
110 views

Newton-Raphson with Zeroth-Order Continuation is not Converging

I am trying to solve this system of nonlinear equations using Newton-Raphson method with continuation (zeroth-order continuation). $F_1(x, y, \xi, \nu) = \left(1 - \frac{1}{\nu}\right) \left(1 - x\...
Abdeljalil's user avatar
2 votes
0 answers
58 views

Newton method not converging with square root equations

I am applying Newton's method to solve the following nonlinear equation systems $F = 0$: \begin{equation} F = \left[\begin{array}{l} \sqrt{(x_0 - x_4)^2 + (x_1 - x_5)^2} - 6 \\ \sqrt{(x_0 - x_2)^2 +...
Pew's user avatar
  • 121
0 votes
0 answers
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PETSc non-linear solvers (SNES): specifying single Eval & Jacobian function

The PETSc documentation example of a non-linear solver call has the user provide separate functions for the Jacobian and function evaluations: ...
Sardine's user avatar
  • 378
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0 answers
117 views

Algorithm to solve system of nonlinear 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 ...
Andres's user avatar
  • 1
0 votes
2 answers
216 views

How and when to impose inhomogeneous Dirichlet boundary conditions in Newton method solver for a PDE?

I am applying a central Finite Difference scheme in space and an implicit Euler scheme in time on a variant of the 2d Burgers equation, of the form:$$u_t + uu_x + uu_y = \nu(u_{xx} + u_{yy})$$ where $...
Robby Ram's user avatar
1 vote
1 answer
139 views

Which analogs of Newton's multivariate method are faster?

Currently, I am studying a 2D nonlinear Schroedinger equation and searching for the fastest method. $$ \begin{equation} i \frac{\partial \psi}{\partial t} = [-\frac{1}{2} \nabla^2 + V_0(r) - i\gamma ...
Andrew's user avatar
  • 31
3 votes
0 answers
101 views

A way to solve nonsmooth stiff ODEs

Let us considered the following ODEs \begin{align*} \dfrac{dX}{dt} = F(X), \tag{1.1} \end{align*} where the unknown $X\in \mathcal{D} \subset \mathbb{R}^l$ and it can be stiff. These ODEs can be ...
Tung Nguyen's user avatar
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0 answers
71 views

accelerating solutions of ODEs with close by parameters

Suppose $\mathbf{u}^1\in\mathbb{R}^n$ is an unknown and $\mu^1\in\mathbb{R}$ is a known parameter. Suppose we have solved the non-linear system of equation for $\mathbf{u}^1$ using Newton's iterations....
NNN's user avatar
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8 votes
2 answers
2k views

Are Quasi-Newton methods computationally impractical?

I was reading a book by Simon Haykin on neural networks when I came across the following strong statement (on the pdf's page 188): "However, we still have a computational complexity that is $\...
Rubem Pacelli's user avatar
0 votes
0 answers
155 views

Discontinuous Galerkin failing to converge Euler equations under p-refinement

I am solving the steady state compressible Euler equations in conservation form in 2D in a rectangular domain with a discontinuous Galerkin (DG) code I have developed. As a test, the boundary ...
Wil's user avatar
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1 vote
0 answers
58 views

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 ...
Aleksandr Borisov's user avatar
4 votes
2 answers
287 views

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 ...
underdog's user avatar
3 votes
1 answer
144 views

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 ...
Bruce Lee Jun Fan's user avatar
1 vote
0 answers
68 views

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(...
Pedro Secchi's user avatar
7 votes
1 answer
207 views

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 ...
Daniel Shapero's user avatar
1 vote
0 answers
63 views

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 ...
Greg Hammett's user avatar
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0 answers
89 views

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}...
Daneel Olivaw's user avatar
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0 answers
85 views

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 ...
kittycat's user avatar
4 votes
3 answers
2k views

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://...
Camille C's user avatar
1 vote
1 answer
719 views

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 ...
Mathieu Rousseau's user avatar
2 votes
2 answers
239 views

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 ...
user37371's user avatar
1 vote
0 answers
72 views

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 ...
Chenna K's user avatar
  • 944
0 votes
1 answer
236 views

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 ...
user37077's user avatar
2 votes
1 answer
224 views

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)\...
vlovero's user avatar
  • 63
1 vote
0 answers
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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 ...
Charlie S's user avatar
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2 votes
0 answers
62 views

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 ...
RockedSalad121's user avatar
0 votes
1 answer
38 views

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 = -\...
RockedSalad121's user avatar
2 votes
1 answer
1k views

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-...
NateM's user avatar
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1 vote
0 answers
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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 ...
K.Cl's user avatar
  • 149
5 votes
1 answer
252 views

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 (...
Abel Thayil's user avatar
3 votes
1 answer
1k views

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 ...
bruin's user avatar
  • 133
2 votes
1 answer
273 views

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 ...
HD189733b's user avatar
  • 121
1 vote
2 answers
945 views

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 ...
Gennaro Arguzzi's user avatar
2 votes
1 answer
592 views

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 ...
arc_lupus's user avatar
  • 553
2 votes
1 answer
507 views

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 ...
Rui Martins's user avatar
1 vote
0 answers
380 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) ...
ZelelB's user avatar
  • 131
2 votes
1 answer
426 views

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]-\...
Nothingts's user avatar
2 votes
1 answer
1k 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 ...
Rasmus's user avatar
  • 95
2 votes
1 answer
281 views

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)\...
arc_lupus's user avatar
  • 553
6 votes
2 answers
143 views

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\...
Justin Solomon's user avatar
1 vote
1 answer
115 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}\\...
D.Dog's user avatar
  • 11
0 votes
1 answer
125 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-...
arc_lupus's user avatar
  • 553
7 votes
1 answer
687 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-...
computational_scientist's user avatar
2 votes
0 answers
117 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{...
Christoph's user avatar
  • 186
1 vote
2 answers
734 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)...
arc_lupus's user avatar
  • 553
6 votes
1 answer
939 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 ...
user14717's user avatar
  • 2,155
1 vote
1 answer
319 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 ...
Farid Hasanov's user avatar
3 votes
1 answer
3k views

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 $...
VoB's user avatar
  • 550