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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{equation} \begin{aligned} \frac{\partial \rho_{11\mathbf{k}}}{\partial t} =& ...
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1answer
49 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 ...
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1answer
53 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 ...
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1answer
89 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 ...
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2answers
190 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 ...
2
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2answers
130 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 ...
3
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1answer
101 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 ...
3
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2answers
180 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} = ...
3
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1answer
123 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 ...
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0answers
61 views

Increase convergence of non-linear equations resulting from ODEs

I am trying to solve a set of couple ODEs: $V_l(r) - r W_l(r) - f1(r) W_l' = 0\tag 1$ $r^2 h''_l(r) + f2 r h_l'(r) + f3 h_l(r) - f4 U_l(r) = 0 \tag 2$ $\kappa (U_l + h_l) + V_{l+1} + W_{l+1} = ...
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1answer
225 views

Newton's method in interpolation

I've seen that in Newton's method for interpolating polynomials, the coefficients can be found algorithmically using (in Python-ish): ...
3
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3answers
116 views

Best method to find the zero of a decreasing function numerically

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 ...
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2answers
530 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 ...
3
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1answer
129 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 ...
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0answers
48 views

When IRLS and Newton method coincide?

From Wikipedia One of the advantages of IRLS over linear and convex programming is that it can be used with Gauss–Newton and Levenberg–Marquardt numerical algorithms. For example, in logistic ...
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1answer
107 views

Matlab help related with the scaled newton's iteration method

I apologize if this question doesn't fit for this site. I am new for this site. I need little help with matlab code for the method mentioned in this paper for computing the inverse of the matrix. In ...
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0answers
312 views

Newton Iteration method convergence

I wrote a Python code which solves a second degree nonlinear differential equation using the Newton iteration method. The code converges to a 2-cycle within 50 or so iterations. The cycle only ...
3
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1answer
189 views

BFGS methods for constrained elasticity problems

My dear community, I am wondering why BFGS methods are not so widely used for simulating mechanical problems which heavily still relies on inverting the hessian matrix. I am essentially interested ...
2
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1answer
278 views

Non-linear root finding when the Jacobian is almost singular

I'm trying to solve a system non linear-equations: $$ \frac{\partial K(\mathbf{\lambda})}{\partial \lambda_i} - c_i = 0 $$ for $i = 1, \dots, 15$, using Newton's method: $$ \lambda^{k + 1} = \lambda^k ...
3
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2answers
119 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 ...
3
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2answers
274 views

Backward Euler method

Can you explain me how does the backward Euler method works? I have seen the formula and try to understand the method, but what I can't understand is why and how to use the Newton-Rapson method. Do ...
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2answers
742 views

Strategies for Newton's Method when the Jacobian at the solution is singular

I'm trying to solve the following system of equations for the variables $P,x_1$ and $x_2$ (all else are constants): $$\frac{A(1-P)}{2}-k_1x_1=0 \\ \frac{AP}{2}-k_2x_2=0 \\ ...
4
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1answer
2k 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 ...