Questions tagged [newton-method]

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

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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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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 ...
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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) ...
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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]-\...
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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 ...
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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)\...
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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\...
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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 ...
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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}\\...
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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 : $\dot x(t) = f(t,x(t),\lambda)$ ...
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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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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-...
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52 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{...
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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)$...
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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 ...
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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 ...
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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 $...
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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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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 ...
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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 &...
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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 ...
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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 ...
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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 ...
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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, ...
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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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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 ...
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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 ...
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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 $$ \int a(u)\nabla\ u\cdot \nabla v = \int f v $$ ...
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228 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 ...
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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 ...
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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}^{...
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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 ...
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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\...
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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: $$R = A(u_{i}) f(u_{i}) - b $$ $$...
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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 ...
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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<...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 \...
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91 views

Quasi Newton taking very small steps

I have implemented a Quasi-Newton method based on the Hessian approximation. I am noticing that the algorithm takes too many iterations to converge, even though it does converge. What I am not able to ...
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1answer
95 views

The linear system in Quasi Newton method

I have implemented a Quasi Newton method for my problem, where I use the Hessian matrix approximation based approach. Hence, there is a linear system solve in every iteration. I solve the linear ...
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112 views

Solving a large system of nonlinear equations, where timeseries are the unknown

I am trying to solve a problem, which I find quite hard, like, headache-hard. I have to solve the following set of $M$ nonlinear equations: $$F(X)=\begin{bmatrix}f_1 (X)\\f_2 (X)\\...\\f_M (X)\\ \end{...
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1answer
102 views

Are there special methods for solving $f'(z)=0$ for analytic $f$?

I am trying to solve a bunch of equations for the zeros of the derivative of an analytic function, and I would like to know if there exist methods that exploit this structure to provide better ...
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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 ...
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244 views

Quasi Newton method for block diagonal Hessian

I am having an unconstrained optimization problem, where the Hessian is positive semidefinite and block diagonal. The function is strictly convex, hence, the curvature condition ($ s^{T}_{k}y_{k} > ...
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292 views

Applying Newton-Raphson method to system of two differential equations, one time independent, one time dependent

The goal is to couple system of two nonlinear differential equations by applying appropriate space and time discretization and Newton-Raphson scheme. The equations system is $$\left[ \begin{array}{c} ...