Questions tagged [newton-method]

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

Filter by
Sorted by
Tagged with
1
vote
1answer
47 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 ...
1
vote
1answer
91 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]-\...
2
votes
1answer
67 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)\...
1
vote
0answers
66 views

How to the determine the initial conditions of the following coupled non-linear ODEs

I am trying to determine the roots (initial conditions) of $θ'$ and $f''$ in the set of ODEs below so I can solve as an initial value problem using the Runge-Kutta method. I tried using Newton-Raphson ...
6
votes
2answers
263 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 $$ \int a(u)\nabla\ u\cdot \nabla v = \int f v $$ ...
1
vote
0answers
57 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) ...
2
votes
1answer
79 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 ...
0
votes
0answers
32 views

Inverse kinematics BFGS divergence

I am trying to implement inverse kinematics solver using BFGS as stated in the paper Xia2017. In the test experiment, i created 4 objects in 3-dimensional space: Node, Node1, Node2, Node3. Each Node ...
6
votes
2answers
118 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\...
1
vote
1answer
53 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
0answers
23 views

Non linear Parametric BVP with inequalities

Consider a non linear ode in dimension $10$: $\dot x = f(t,x,\lambda)$ where $\lambda$ is a vector of $p$ parameters. Consider a boundary value problem of the form : $\dot x(t) = f(t,x(t),\lambda)$ ...
7
votes
1answer
197 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-...
0
votes
1answer
58 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-...
5
votes
1answer
139 views

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
votes
0answers
179 views

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{...
2
votes
0answers
53 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{...
3
votes
2answers
247 views

System of quadratic algebraic equations

I have this problem $H_i(x_1,x_2,\dots, x_N) = a_{ijk} x_j x_k + b_{ij} x_j + c_i = 0 \quad 1\leq i \leq N$ And I need to show that applying Newton-Raphson can fail to find even one real solution ...
0
votes
1answer
70 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)$...
1
vote
2answers
176 views

Finding null vectors of a parameter-dependent matrix

I have dense complex matrices $M(z)$ in which each element $M_{ij} = M_{ij}(z)$ depends on a complex parameter $z$. I need to find $z$ such that the matrix $M$ gets singular, i.e. I am looking for ...
6
votes
1answer
159 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
1answer
106 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
1answer
193 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 $...
1
vote
1answer
88 views

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 ...
1
vote
1answer
73 views

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 &...
1
vote
1answer
43 views

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 ...
3
votes
0answers
113 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 ...
2
votes
0answers
105 views

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 ...
9
votes
3answers
622 views

Methods of solving non-linear advection-diffusion systems beyond Newton-Raphson?

I'm working on a project where I have two adv-diff coupled domains through their respective source terms (one domain adds mass, the other subtracts mass). For brevity, I'm modeling them in steady ...
2
votes
0answers
130 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, ...
1
vote
2answers
2k views

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, ...
5
votes
1answer
120 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 ...
0
votes
2answers
131 views

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 ...
1
vote
0answers
229 views

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 ...
1
vote
0answers
211 views

Newton - Raphson method : maxima of function in 2 variables

I am computing the maximum of a function (with two-variables) using Newton-Raphson method. The function is : $e^{-(x \ - x_0)^2 - (y \ - y_0)^2}$, whose maxima exists at $(x_0,y_0)$. The Jacobian ...
4
votes
3answers
179 views

How can I use Projected Gradient Descent for this optimization problem with constraint?

Suppose $ q $ and $ A $ are given and that $ q, p \in R^{N} $ and $ A \in R^{NxN} $, then how can I find the vector $ p $ such that $$ (q - p)^{T}A(q - p) $$ is minimized constraint to $ \sum_{i=1}^{...
5
votes
1answer
2k views

Difference between Gauss-Newton method and quasi-Newton method for optimization

Can anybody help me? I heard that Gauss-Newton method compute an aproximation of the Hessian instead of the true Hessian, but, quasi-Newton method too, don't it? what is the differences between them? ...
1
vote
0answers
72 views

Successive iteration method for solving eigenvalue ploblem

I have a question concerning the branch of successive iteration methods (Newton, Runge-Kutta). I definitely know (or can read in Wikipedia) the implementation of these methods. But I was wondering ...
1
vote
0answers
104 views

Jacobian of the electron and hole drift diffusion equations with respect to potential in semiconductors

These are the discretized drift diffusion equations as taken from the book "Analysis and Simulation of Semiconductor Devices". The electron continuity equation is: $$((\Delta_{j-1}^y+\Delta_j^y)/2\...
5
votes
1answer
825 views

Newton's method with box-constraints

I have to use an iterative method (Newton-Raphson, modified Newton and Broyden) to solve a system of nonlinear equations $f(x)=0$. Every unknown $x_i$ is bounded between $l_i$ and $u_i$, i.e., $l_i<...
2
votes
1answer
116 views

Newton's method stagnates at small error

I have a system of the form $$A(u)f(u)=b$$ where $A$ is basically a matrix originating from the Finite Element Method. I try to solve it using the Newton method: $$R = A(u_{i}) f(u_{i}) - b $$ $$...
1
vote
3answers
5k views

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 ...
1
vote
0answers
81 views

Using Newton-Raphson method to solve the hydrostatic equation

I'm trying to use newton-raphson method for nonlinear systems of equations as described in 'Numerical recipies' book in chapter 9.6 to solve the hydrostatic equation for a polytropic star. For each ...
6
votes
1answer
132 views

Roots of a function for eigensystem

I want to find the roots for $\kappa$ for the equation $$\sqrt{\alpha - 1} \cos{\left (\frac{\sqrt{2} \sqrt{\alpha - 1}}{2 \sqrt{\epsilon}} \right )} \cosh{\left (\frac{\sqrt{2} \sqrt{\alpha + 1}}{2 \...
7
votes
3answers
362 views

Strong coupling of a non-linear multiphysic problem: failure with Newton Raphson method

I am trying to solve a multiphysic problem using finite elements and a Newton Raphson solution scheme. I have two non-linear subsystems that are coupled bi-directionally. The first subsystem includes ...
2
votes
0answers
94 views

Optimization based integration for MPM

I'm considering implementing (just for simplicity) the unconstrained implicit optimization based integration for Material Point Method as described in Chenfanfu Jiang's thesis on MPM (the minimization ...
2
votes
0answers
163 views

Does applying the Newton-Raphson iteration for matrix reciprocal refine a matrix inverse from LU/GE?

This is a follow-up to this answer. Suppose you have a possibly very ill-conditioned matrix $A$, and you compute its inverse with LU/GE to get $X_{\text{lu}}\approx A^{-1}$. The Newton-Raphson ...
6
votes
1answer
179 views

What would be a good approach to solving this large data non-linear least squares optimisation

Introduction to Problem I'm using a Truncated Signed Distance Function to perform 3D reconstruction from depth images. Essentially I have a large voxel grid where each voxel contains the signed ...
3
votes
2answers
454 views

Low-rank updates in BFGS

I have read this and other threads on this site on BFGS, but I still don't have a clear understanding of what it's meant by low-rank updates. For example, I read the following in this book: The ...
2
votes
1answer
59 views

Doubt regarding principled approach towards approximating the Hessian

In my optimization problem, the hessian has a structure such that it can be written as the sum of two matrices. Populating the first of the matrices is efficient. Populating the second one is ...
7
votes
1answer
290 views

Eigenvalues of $ab^T$

In deriving a Newton scheme, I end up with a Jacobian matrix of the form $J=I+ab^T$ where $a,b$ are vectors. For practical reasons, I want to approximate it by a symmetric positive definite matrix. ...