Questions tagged [nonlinear-equations]

Solution of nonlinear systems of equations. The equations might be algebraic or differential equations.

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16
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1answer
4k views

When is Newton-Krylov not an appropriate solver?

Recently I have been comparing different non-linear solvers from scipy and was particularly impressed with the Newton-Krylov example in the Scipy Cookbook in which they solve a second order ...
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2answers
2k views

Is it possible to solve nonlinear PDEs without using Newton-Raphson iteration?

I am trying to understand some results and would appreciate some general comments on tackling nonlinear problems. Fisher's equation (a nonlinear reaction-diffusion PDE), $$ u_t = du_{xx} + \beta u ...
13
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1answer
3k views

Comparing Jacobi and Gauss-Seidel methods for nonlinear iterations

It is well known that for certain linear systems Jacobi and Gauss-Seidel iterative methods have the same convergence behavior, e.g. Stein-Rosenberg Theorem. I am wondering if similar results exist for ...
11
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2answers
506 views

Numerical method for equation solving that works on stochastically computed functions

There are many well known numerical methods for solving equations of the type $$ f(x) = 0, \quad x \in \mathbb{R}^n,$$ e.g. bisection method, Newton's method, etc. In my application $f(x)$ is ...
10
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3answers
4k views

Solution of quartic equation

Is there a open C-implementation for the solution of quartic equations: $$ax⁴+bx³+cx²+dx+e=0$$ I am thinking of an implementation of Ferrari's solution. On Wikipedia I read that the solution is ...
10
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1answer
321 views

Solving a difficult system of equations numerically

I have a system of $n$ non-linear equations that I want to solve numerically: $$\mathbf{f}(\mathbf{x})=\mathbf{a}$$ $$\mathbf{f}=(f_1,\dots,f_n)\quad\mathbf{x}=(x_1,\dots,x_n)$$ This system has a ...
9
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3answers
1k views

Basin of attraction for Newton's method

Newton's method for solving nonlinear equations is known to converge quadratically when the starting guess is "sufficiently close" to the solution. What is "sufficiently close"? Is there literature ...
9
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3answers
728 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 ...
9
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2answers
1k views

What does the Von Neumann's stability analysis tell us about non-linear finite difference equations?

I am reading a paper [1] where they solve the following non-linear equation \begin{equation} u_t + u_x + uu_x - u_{xxt} = 0 \end{equation} using finite difference methods. They also analyse the ...
8
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5answers
366 views

Iterative solution to a nonlinear equation

I appologize in advance if this question is silly. I need to compute the root of \begin{equation} u -f(u) =0 \end{equation} Where $u$ is a real vector and $f(u)$ is a real-vector valued function. ...
8
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2answers
2k views

Is the shooting method the only general numerical method for solving nonlinear boundary value ODEs?

During my wandering in Mathematica.se, I gradually noticed that a certain kind of differential equation solving problem is "troubling" us all the time, that is, the boundary value problem (BVP) of ...
8
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1answer
532 views

Nonlinear wave equation - Finite element or finite difference

I would like to know the which is more advantageous when it comes to solving nonlinear hyperbolic equations, Finite Element or Finite difference methods? Which method will be better in capturing ...
8
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1answer
741 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} -...
8
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1answer
143 views

F(x) = 0 vs. ||F(x)||^2->min

In many areas of application, one needs to solve a nonlinear system of equations $$ F(x) = 0. $$ Sometimes, the formulation $$ \|F(x)\|^2 \to\min $$ is used. Clearly, every solution $\hat{x}$ of $F(x)=...
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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{...
8
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0answers
112 views

Review of modern homotopy methods and practical techniques

I'm hoping someone can recommend recent literature concerning homotopy methods for solving systems of nonlinear equations. Already by the time of Layne Watson's 1986 paper there were a lot of methods,...
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4answers
7k views

How to find more than one root of a polynomial?

This program finds the first root of the function f, defined in the code. There are 5 roots of this function. (x=1,2,3,4,5) I wish to find all of the roots in this program and print them to the screen....
7
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2answers
168 views

For a non-linear PDEs should the source term be discretised at $u_j$ or averaged over $(u_{j+1} + u_{j-1})/2$?

The non-linear Poisson equation in one-dimension, $$ 0 = \frac{\partial^2u}{\partial x^2} - f(u) $$ can be discretised as to give, $$ u_{j-1} -2u_{j} + u_{j+1} = h^2 f(u_j) $$ where $h$ is the ...
7
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3answers
387 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 ...
7
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1answer
118 views

Understanding the Eisenstat-Walker method for choosing the tolerance of a linear solver when solving a non-linear PDE

We are working on the solution of large non-linear PDE (say the Navier-Stokes equation) which we solve using Newton's method with an analytical formulation of the jacobian. For very large systems, we ...
7
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1answer
428 views

Numerical method for nonlinear system of algebraic equations of special type

I have a nonlinear system of algebraic equations of special kind: $$ \begin{array}{rcl} x_{i}+y_{i}+z_{0,1}+c_{i,1}z_{1,1} & = & d_{i,1}, \\ x_{i}^2 + y_{i}^2 + z_{0,2} + c_{i,1} z_{...
6
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3answers
1k views

Convergence of fixed point iterations of a non-linear matrix system

I'm working on modeling two phase immiscible flow in a porous medium. When I setup the system of equations, I obtain a non-linear system of equations that can be expressed in the form: $A(x)x=b$ ...
6
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2answers
5k views

Solve non-linear set of three equations using scipy

I need to solve a non-linear set of three equations using scipy. However, I do not have any clue on which algorithm is suitable for my problem from a mathematical point of view (stability, ...
6
votes
1answer
191 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 ...
6
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4answers
2k views

parameters estimation

I have to estimate a parameter (K), but I don't know how I can do it. I think by a regression model (minimum least square?), but I'm not sure. The system is: ...
6
votes
1answer
879 views

Algorithm for solving system of quadratic equations and linear equations

Let $x \in R^N$. From a Spectral Chebyshev collocation method, I have a system of quadratic and linear equations. Denote them, $$ x^T Q_i x + L_i^T x = 0 $$ and $$ A x = 0 $$ Furthermore, I know ...
6
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4answers
750 views

Large-scale nonlinear optimization problem

I want to solve a nonlinear optimization problem of the following form \begin{equation} \min\left(\sum_i d^{x_i}c_{i}\right)\\ 0 \leq x_{i} \leq a\\ \sum_{i} x_{i} \leq b \end{equation} $a$, $b$, $...
6
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1answer
155 views

Numerical methods for boundary-value ODEs with a jump condition

I want to solve a non linear system of equations of a particular kind. I find it hard to formulate clearly so I directly give a simple example. $ f''=A(f,g)\\ g''=B(f,g) $ with the boundary ...
6
votes
1answer
306 views

Non-linear optimization using approximate gradient

I'm working with non-linear optimization for imaging, such as MRI and CT. Our problem is of the form $\|Ax-b \|_2^2+\lambda \|Wx\|_1$. $A$ is never formed explicitly, so we're limited to approaches ...
6
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1answer
817 views

Nonlinear dynamics: algorithm suggest

I've just started a thesis on nonlinear dynamics which entails numerical analysis of the Duffing oscillator (DO). It's basically just a second order ODE, or equivalently a set of ODEs. Say, after ...
6
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0answers
496 views

Implementing a Hill-Type Muscle Model

I'm interested in implementing the muscle model used in Geijtenbeek and Wang et al's work. Both papers link to the paper by Geyer and Herr, which describes this model: However, the paper on this ...
5
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2answers
306 views

Implementation of nonlinear term in FEM

Although there are similar questions, I am also struggling with the implementation of the following term in "my own code" by Finite Element Method, namely, $\nabla \phi \cdot \nabla \phi$. $\phi$ is ...
5
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2answers
182 views

What are the numerical methods for huge polynomial systems?

Let a system of $n$ polynomial equations of degree $d$ with $m$ variables. I'm interested in a sparse system with $d = 3$, $n \sim 2000000$, $m \sim 50000$ and integer coefficients. What techniques ...
5
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2answers
273 views

How to do upwinding in finite volume schemes for nonlinear equations?

In finite difference theory, you learn, that you have to use upwinding for equations with high convection, like Burgers' equation. What does the finite volume equivalent look like? What if the ...
5
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1answer
1k views

Confusion regarding the Adam-Moulton and Backwards Differentiation Formula (BDF) of the VODE solver

I am exploring the Method of Lines as a way of time stepping semi-discretised PDEs with ODE time-integration solvers. For an excellent introduction to this technique see the scholarpedia.org article. ...
5
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1answer
6k views

Newton-Raphson method for nonlinear partial differential equations

For the numerical solution of Reynolds equations (a non-linear partial differential equation), the Newton-Raphson method is generally proposed. After getting algebraic equations from a finite ...
5
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1answer
101 views

What are systematic ways of approximating a non-smooth (non-continuously differentiable) system dynamic to be n-smooth?

I have a system dynamic that is non-smooth because it has several signum and absolute value functions in it (three-tank level control). I can obviously choose different sigmoid functions to ...
5
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2answers
152 views

Convergence issues for a non-linear system

I have a nasty system of coupled integral equations, which I managed to discretize and recast a non-linear system, i.e. something like: $$ \vec{w} = F \left( \vec{w} \right) \hspace{32pt} w \in \...
5
votes
1answer
127 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 ...
5
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1answer
165 views

Quantifying the degree of nonlinearity in a heat transfer problem

I’m working with a heat equation of the form. $$\frac{d(\rho(T)c_p(T)T)}{dt}-\nabla\cdot(k(T)\nabla T)=f$$ with temperature dependent density $\rho(T)$, specific heat $c_p(T)$, and thermal ...
5
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1answer
144 views

Evolving nonlinear Schrodinger equation with higher-order algorithms?

First I will give the relevant information for my question, and then I'll ask the question. $\large{\textrm{Background}}$ For evolving the nonlinear Schrodinger equation (NLS), one typically uses [a ...
5
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1answer
249 views

Eikonal Equation solver with different grid densities

The Fast Marching Method, Fast Iterative Method, and Fast Sweeping Method are three ways of solving the Eikonal Equation on a discrete grid, essentially just a wavefront spreading out from initial ...
5
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2answers
199 views

Solving a nonlinear algebraic system that includes a linear term

I am trying to solve a particular system of non linear equations written as $F(x) = 0$ in an efficient way. More specifically, $$F(x) = (I - \gamma A)x - g(x) + C$$ where $\gamma$ is a scalarconstant,...
5
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1answer
262 views

Convergence of Jacobi's method for a semilinear elliptic PDE

I have an iterative finite difference scheme for the Poisson equation $$ \nabla^2 u=-\rho $$ It's the Jacobi method, which has the form (for 1D systems) $$ u^{n+1}_{i} = \frac{1}{2}(u^n_{i+1} + u^n_{...
5
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1answer
502 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{...
5
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0answers
47 views

2nd-order TVD criteria for flux-limiter

Consider a nonlinear hyperbolic conservation equation: $$ \partial_{t}u = -\partial_{x}f(u) $$ The spatial derivative of $f(u)$ may approximated after a spatial discretization by $x_{j}=j\Delta x$ $$ \...
4
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2answers
197 views

treating “almost linear” nonlinear least-squares problems

As a model for a nonlinear least-squares problem with a large linear part problem, consider $$ \Delta u = 0 \quad\text{in } \Omega,\\ n\cdot\nabla u = 0 \quad \text{on } \Gamma,\\ (u(x_i) - u(y_i))^2 =...
4
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1answer
713 views

Should the Jacobian of a system of PDEs be calculated from the main equations of the discretised equation?

I am solving a coupled system of non-linear PDEs in 1D. Something like, $$ u_t = F_1(u,v,w) \\ v_t = F_2(u,v,w) \\ w_t = F_3(u,v,w) $$ where each variable is a function of $x$ (the spatial dimension)...
4
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1answer
621 views

Stability of numerical schemes for non-linear equations with a Jacobian with negative eigenvalues

Let us assume I have an A-stable numerical scheme. I believe that given any linear equation $y' = Ay$, it means that the numerical scheme applied to this equation is stable (and therefore convergent ...
4
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1answer
133 views

What is the preferred method for evolving the Nonlinear Schrödinger Equation?

I am interested in evolving the (cubic) self-focusing nonlinear Schrödinger equation, $$i\frac{\partial \psi}{\partial t}+\frac{1}{2}\frac{\partial^2 \psi}{\partial x^2}+\left|\psi\right|^2\psi=0$$ ...

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