Questions tagged [newton-method]
Method for finding successively better approximations to the roots (or zeroes) of a real-valued function
109
questions
0
votes
1answer
26 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 $$\nabla^2 f p_k = -\...
2
votes
1answer
102 views
Constrained Newton-Raphson root finding
Original Question
I have a set of non-linear equations and I need to find the root where a subset of my solution vector is constrained to be greater than or equal to 0. I have implemented the Newton-...
0
votes
0answers
22 views
Solving natural convection equations (heat and flow) with the shooting method (managing 2 boundary values)
TL;DR I've been implementing a python program to solve numerically equations for natural convection based on a particular similarity variable using runge-kutta 4 and the shooting method. However I don'...
1
vote
0answers
39 views
Inverse Newton Method for optimization: is this the correct algorithm?
I am trying to implement the algorithm in this article. I have already asked a question before about it here, and I am trying to figure out what I am doing wrong. This time, it's this section of the ...
5
votes
1answer
80 views
How do I globally change the precision of a piece of code in Python to debug it?
I am solving a system of non-linear equations using the Newton-Raphson method in Python. This involves using the solve(Ax,b) function (...
0
votes
0answers
116 views
Jacobian-free approach for time-dependent equations for implicit time-stepping
When solving time-dependent non-linear equations, such as the non-linear diffusion equation
$$\partial_tu=\nabla\left(D(u)\nabla u\right)$$
usually Newton's method is applied, with (coupled with the ...
3
votes
1answer
161 views
Parallelizing Newton-method in solving non-linear systems
Circuit simulation software based on SPICE (such as ngspice) uses Newton-Raphson method to solve non-linear system of equations ...
1
vote
0answers
52 views
How to avoid gsl root finder evaluate function outside its domain
When I use the newton's method or hybrid solver in the GSL package to deal with 1-D or multidimensional root solving problems, the code frequently crashes when the solver requests function value ...
1
vote
2answers
127 views
Evaluation of slope at iteration ith - Newton-Raphson method
I'd like to know how Ansys computes the slope (=stiffness matrix) at point x1 in figure. I'm studying the way in which Ansys uses the Newton-Raphson method when there are nonlinearities.
In the slide ...
2
votes
1answer
126 views
Number of GMRES iterations increase when stepping forward in time, using the Newton method
I am solving a system of nonlinear time-dependent equations using the Newton method in a finite-element-setting, i.e. first I create the jacobian matrix for the current time, and afterwards I try to ...
2
votes
1answer
169 views
How to use Newton-Raphson method to handle nonlinear terms in coupled system of PDEs?
I'm trying to solve the Nonlinear Schrodinger's Equation (NLSE) in 2D using Finite Elements, but I don't know how to handle the nonlinear term. I suppose I have to apply the Newton-Raphson algortihm ...
1
vote
0answers
78 views
How to use Wolfe-Powell step-size control in quasi-Newton method?
I'm trying to find the minimum of a function using the quasi-Newton method with the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm.
But I want to change the following implementation, so that:
1) ...
2
votes
1answer
177 views
Solving nonlinear PDE with finite difference based on Newton-Krylov
I am now working on solving MHD equations with finite difference method, which include nonlinear equations:
$$
\frac{\partial\rho}{\partial t}+\nabla\cdot\left[\left(\rho_0+\rho\right){v}\right]-\...
2
votes
1answer
160 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 ...
2
votes
1answer
96 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
$$\begin{split}
c_A(A, B)\partial_tA&=\nabla\left(k_A(A, B)\nabla A\right) + f_A(A, B, t)\\
c_B(A, B)\...
6
votes
2answers
125 views
Positive root of $x^q + bx - b$
Is there either a closed-form expression or fast/elegant algorithm for computing the positive root of the polynomial
$$f(x)=x^q + \beta x - \beta,$$
where $\beta>0$ and $q\geq2$? How about the $q\...
1
vote
1answer
72 views
Product of rank one updates as a low rank update for quasi newton/BFGS
I'm trying to improve the speed of the following iteration to calculate $s_k$:
$$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
1answer
71 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-...
7
votes
1answer
245 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-...
2
votes
0answers
73 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{...
1
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2answers
246 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
$$J(u)\delta u=-F(u)...
6
votes
1answer
287 views
Defining a condition number and termination criteria for Newton's method
The condition number of function evaluation
$$
\mathrm{cond}(f,x) := \left| \frac{x f'(x)}{f(x)} \right|
$$
is infinite at a root of $f$. Hence it is useless for rescaling a tolerance which defines an ...
1
vote
1answer
162 views
Shooting method implementation
I have a trouble of defining a Jacobian matrix for my problem. Basically, I have 4 differential equations to be solved.
$$
\begin{aligned}
\dot x_1(t)&=x_2(t)\\
\dot x_2(t)&=p_2(t)ā\sqrt 2 ...
3
votes
1answer
397 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 $...
8
votes
0answers
184 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{...
1
vote
1answer
101 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
95 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
79 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
158 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
2answers
518 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 ...
2
votes
0answers
203 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
4k 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
125 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
154 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 ...
7
votes
2answers
353 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
370 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
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0answers
475 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
301 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}^{...
1
vote
0answers
75 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
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0answers
128 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:
$$((\Delta_{j-1}^y+\Delta_j^y)/2\...
2
votes
1answer
127 views
Newton's method stagnates at small error
I have a system of the form
$$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
0answers
85 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 ...
5
votes
1answer
1k 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
0answers
101 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
288 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
190 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
695 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
62 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
323 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. ...
6
votes
1answer
149 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 \...
