# Tagged Questions

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

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### 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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### What does a negative time stepping mean? (Adaptive time stepping)

Summary behind the problem: The following code aims at solving a static elasto-plastic problem. Like a 2D square mesh based on an elasto-plastic constitutive model like Von-Mises or Drucker-Prager ...
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### 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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### 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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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} > ... 0answers 33 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 $$\left[ \begin{array}{c} ... 2answers 103 views ### Why do Newton-Krylov iterations stagnate in this problem? [closed] Consider this integro-differential heat equation taken from SciPy documentation page: \nabla^2 P = \alpha \left(\iint_\Omega \cosh(P)dx dy \right)^2 which was found in this question. In the ... 1answer 54 views ### Efficient and stable computation of inverse CDF What is the most efficient and numerically stable algorithm for computing the inverse CDF F^{-1}(y) of a probability function, assuming that both the PDF f(x) and the CDF F(x) are known ... 1answer 175 views ### Solving a set of linear equations with block structure and weak coupling I have a standard set of linear equations Ax=b where the Hessian matrix A has the special block structure as shown: A= \begin{pmatrix} T & U\\ U^T & V \end{pmatrix}, x= \begin{... 1answer 54 views ### Implementation of Backward-Euler scheme, Newton-Raphson iteration scheme to time dependent nonlinear differential equation I just knew how to do Newton-Raphson iteration in time-independent 1D nonlinear differential equation. Then I applied to time-dependent 1D nonlinear differential equation, and I got confused. Below ... 0answers 60 views ### Best way to add a positivity constraint to Newton's Method So given an objective function f({\bf x}), I would like to include a positivity constraint when I perform the fixed point iteration:$${\bf x}^{(t+1)}={\bf x}^{(t)} - \text{H}_f^{-1}\nabla f({\bf x}^... 2answers 113 views ### How to implement Newton method in solving 1D PDE system? (ie. Poisson eq, continuity eq, drift-diffusion eq.) I want to solve PDE system, which consists of Poisson equation, continuity equations for electron and hole with drift-diffusion equation numerically, by using method called Newton's method. This ... 1answer 149 views ### Newton's method goes to zero determinant Jacobian I am using the Newton's method to solve$3\times3$systems. For some particular cases, it turns out that at a given iteration, the Jacobian matrix cannot be inverted and that its determinant is very ... 1answer 409 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? ... 1answer 79 views ### Solving this nonlinear system of equations Suppose I have this set of equations: $$a = x + z\qquad (1)$$ $$b = y + \frac{z}{2}\qquad (2)$$ $$z = k_0x\sqrt{y}\qquad (3)$$ Where$a$,$b\in \mathbb{R}$and$k_0 > 0$. The values of$a... 1answer 151 views ### Newton iteration applied to nonlinear PDE I'm having difficulty understanding how to apply Newton iteration to nonlinear PDEs and then use a fully implicit scheme to time step. For example, I want to solve Burgers equation $$u_{t} + u u_{x} -... 1answer 89 views ### Inverse Transformation of Isoparametric Quadrilaterals I urgently need help with a problem I am having for an assignment for university. I do not expect a full solution, but I am really in need of guidance. I am required to write a FEA program in VBA, I ... 2answers 149 views ### Ill-conditioned Jacobian matrix from Nernst-Planck equation with Butler-Volmer reactions The governing equations are listed here of my notes on page 4. It's a reproduction of other's paper which solves the equations with COMSOL. The problems arise when I want to solve for the consistent ... 0answers 47 views ### Monotonic convergence of Newton's method for boundary value problems I’m interested in solving nonlinear elliptic boundary value problems of the type$$ -a\Delta u + f(u) = 0,  u|_\Gamma = u_0 by Newton’s method when its convergence is global and monotonic. ... 0answers 175 views ### Linearization of two phase flow iteration for Newton Method I am trying to implement a numerical model to solve a two-phase flow using the Newton-Raphson method. To do so, I also have to differentiate the fluid pressure P with respect to the matrix velocity ... 1answer 145 views ### scaling and preconditioning for trust region Newton methods Geometrically, scaling and preconditioning seem to address similar challenges in optimization. However, these two concepts are implemented very differently. Take trust region Newton method, as an ... 1answer 197 views ### Can an approximated Jacobian with finite differences cause instability in the Newton method? I have implemented a backward-Euler solver in python 3 (using numpy). For my own convenience and as an exercise, I also wrote a small function that computes a finite difference approximation of the ... 1answer 156 views ### Hessian-free and Truncated Newton methods In this paper on Deep Learning for Machine Learning, the approach is referred to as Hessian-free method. That is because the Hessian is never computed explicitly. Instead, the product of the Hessian ... 1answer 177 views ### Caveats of Hessian free method Hessian free iterative optimization techniques like Newton-CG, do not explicitly compute the Hessian but instead approximate the product of the Hessian with a vector through finite difference. The ... 2answers 223 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 ... 1answer 165 views ### Doubt regarding stopping criterion for Newton method I am solving an unconstrained convex optimization problem, which can easily have a million variables. I am trying to get a working system with a toy problem of around 200 variables. I am noticing that ... 1answer 117 views ### Strict Feasibility in Interior Point Methods As we know, in the interior point methods, all the iterates have to be strictly feasible. I implemented an affine scaling interior point for nonlinear objective functions. For small examples (2D), it ... 0answers 59 views ### Mapping to a computationally less expensive basis when employing Newton's method I'm looking for advice, or references, for a change of basis to my dependent variables that leads to a less computationally expensive scheme when solving a system of coupled polynomial equations. ... 1answer 128 views ### Methods for Constrained Optimization Problems with Box Constraints Consider this problem: $$\begin{array}{ll} \text{minimize } & f(x) \\ \text{subject to } & a \leq x \leq b \end{array}$$ where a,b,x \in \mathbb{... 1answer 163 views ### Active Elements in Projected Newton's Method? To those who are familiar with the projected Newton's method or projected gradient method... We consider a constrained optimization problem with simple bounds. Particularly, minimize f(x) subject to ... 2answers 1k 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 ... 3answers 332 views ### Beale's function and newton iteration I am trying to find the minimum of the so called Beale’s function given by f(x_1,x_2) = (1.5-x_1+x_1x_2)^2 + (2.25-x_1+x_1x_2^2)^2 + (2.625-x_1+x_1x_2^3)^2 Using Newton iteration x^{(k+1)} = x^{... 2answers 372 views ### Do I need to impose boundary conditions in the Jacobian matrix? In the framework of Finite Element Method, when the Newton method is used, we solve J(x^k) \delta x = -f(x^k), and the increment \delta x would not change some entries from x^k related to ... 1answer 348 views ### Solving a system of nonlinear equations with an ODE solver is faster than with the Newton method? This is somehow unexpected, but my recent experience with solving a system of nonlinear equations is that treating them as the right hand side of a system of ordinary equations and then evolve the ... 1answer 44 views ### Non linear system of equations with discretization on k-space I want to numerically solve the following system of differential equations at the steady state: \begin{aligned} \frac{\partial \rho_{11\mathbf{k}}}{\partial t} =& +\frac{i}{\... 1answer 199 views ### MATLAB Newton non-linear equation I have the following non-linear equation: where w0=0.25,w0=0.5,w0=0.75. I have to prove that if k is a root, then also −k is a root and that there exists only one k∈(0,1) root, but my ... 1answer 77 views ### Calculation of isotropic-nematic phase transition In this paper, the theory behind the isotropic-nematic phase transition is discussed. Furthermore, an algorithm is given to calculate some properties of this phase transition. I have written a ... 1answer 122 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 ... 2answers 493 views ### Newton-Raphson method fails! I am trying to solve an equation like R(x) = 0, using Newton-Raphson method. To obtain the x increment in each iteration I solve dx = -(A)^{-1}\cdot R where A = dR/dx. But the convergence ... 2answers 324 views ### Poisson-Nernst-Planck equations with ill-conditioned sparse matrix I am trying to solve Poisson-Nernst-Planck system of equations for ions diffusion problem using finite volume method. Nernst-Planck equation for mass transport and Poisson equation for electrostatic ... 1answer 152 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 ... 2answers 397 views ### Slow convergence of Newton's method for finite elements The application is a simple non-linear advection diffusion problem (steady state) using DGFEM. My error at each iteration is given by e_{n+1} = ||\mathbf{J}^{-1}(\mathbf{u}_{n})\mathbf{F}(\mathbf{u}... 1answer 137 views ### Excluding roots from a system of nonlinear equations I have a system of nonlinear equations of which I know it has a single root I am interested in, and has a continuum of roots I am not interested in. I am currently using Newton with line-searching in ... 0answers 73 views ### Increase convergence of non-linear equations resulting from ODEs I am trying to solve a set of couple ODEs:V_l(r) - r W_l(r) - f1(r) W_l' = 0\tag 1r^2 h''_l(r) + f2 r h_l'(r) + f3 h_l(r) - f4 U_l(r) = 0 \tag 2\kappa (U_l + h_l) + V_{l+1} + W_{l+1} = 0\...
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 higly dependent of ...