Questions tagged [newton-method]
Method for finding successively better approximations to the roots (or zeroes) of a real-valued function
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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\...
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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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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 +...
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Solving Kepler equation for true or eccentric anomaly
Is there any reason to always solve the Kepler equation for the eccentric anomaly, $E$, instead of the more meaningful (at least to me) true anomaly, $\theta$?
Solving for the eccentric anomaly ...
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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:
...
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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 ...
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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 $...
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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 ...
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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 ...
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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....
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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
$\...
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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 ...
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Best method to find the zero of a decreasing function numerically
I need to find the zero of a function $f(\lambda)$ which is of the form $\sum \frac{c_i^2}{(1+\lambda d_i)^2} -1 $. I tried using Newton's method, and it works sometimes, but it is highly dependent on ...
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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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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]-\...
CommunityBot
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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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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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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:
$$
\begin{Bmatrix}
\frac{\partial x}{\partial t} \\
0
\end{Bmatrix} = \begin{Bmatrix}
f(...
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Which absolute and/or relative stopping criteria do use for Newton's method?
I saw many stopping criteria for Newton's method all around Web and books.
Some are defined from the residuals:
of either current iteration only:
$$
\|f(\mathbf{x}^{(k)})\| \leq \epsilon
$$
(https://...
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tiny typo in Numerical Recipes Eq. 9.4.6 [closed]
The Numerical Recipes Forum http://numerical.recipes/forum/ is closed, so I will record a tiny typo here for the benefit of others who may wonder about this. (This typo is not in the software, but ...
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Blown-up iterates in Gauss-Newton method
I am working on a non-linear least squares problem with standard form, in which I need to calibrate a parameter vector $\Theta$ to a set of inputs $\mathbf{x}$ and outputs $\mathbf{y}$:
$$\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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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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votes
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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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)\...
- 2,440
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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 ...
- 2,069
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1
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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 $$\nabla^2 f p_k = -\...
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Constrained Newton-Raphson root finding
Original Question
I have a set of non-linear equations and I need to find the root where a subset of my solution vector is constrained to be greater than or equal to 0. I have implemented the Newton-...
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Inverse Newton Method for optimization: is this the correct algorithm?
I am trying to implement the algorithm in this article. I have already asked a question before about it here, and I am trying to figure out what I am doing wrong. This time, it's this section of the ...
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Which SciPy nonlinear solver when Jacobian is analytically known and sparse?
I have a nonlinear function fun with n inputs and n outputs. I also have a function jac which calculates the Jacobian, which is ...
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votes
1
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How do I globally change the precision of a piece of code in Python to debug it?
I am solving a system of non-linear equations using the Newton-Raphson method in Python. This involves using the solve(Ax,b) function (...
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1
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Residual value goes to NaN while solving a system of nonlinear equations
I am solving a system of coupled nonlinear equations using Newton's method, similar to
$$\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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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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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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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 $s_k$:
$$B_k^{-1} = \Bigg( I + \frac{s_{k}s_{k-1}^T}{||s_{k-1}||^2}\Bigg)...\Bigg(I+ \frac{s_1s_0^T}{||s_0||^2}\Bigg) B_0^{-1}\\...
CommunityBot
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vote
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answers
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Implementation of the Jacobian-free Newton method
In my calculation (of a simple heat equation, for testing) using the Newton method, I tried to replace the full Jacobian matrix with an approximation vector, i.e. replacing $J$ in
$$J(u)\delta u=-F(u)...
- 8,602
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votes
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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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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
$$ \int a(u)\nabla\ u\cdot \nabla v = \int f v $$
...
- 8,602
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0
answers
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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) ...
- 8,602
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votes
1
answer
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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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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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votes
1
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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-...
- 2,135
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votes
1
answer
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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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Matlab help related with the scaled Newton's iteration method
I need a little help with Matlab code for the method mentioned in this paper for computing the inverse of the matrix.
In this paper, the author presented scaled Newton iteration given by $X_{k+1} = \...
- 8,602
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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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