# Questions tagged [newton-method]

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

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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)\...
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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 ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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}^{... 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 ... 0answers 133 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: $$((\Delta_{j-1}^y+\Delta_j^y)/2\... 1answer 131 views ### Newton's method stagnates at small error I have a system of the form$$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 ... 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 ... 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<...
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 ...