# Questions tagged [convergence]

Questions related to whether the sequence of iterates generated by an iterative method has one or more limit points, and if those limit points have the correct properties.

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### Convergence rate Jacobi/Gauss-Seidel with mesh resolution

In the book A Multigrid Tutorial - Briggs, Henson. McCormick in the beginning of Chapter 3, it is mentioned that ...because the convergence factor behaves as 1-$O(h^{2})$, the coarse grid ...
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### CFL condition for variable coefficients

I understand that the constant in the Courant-Friedrichs-Lewy condition is defined as $\mathrm{CFL} = \frac{u \Delta t}{\Delta x}$, where $u$ is the principal coefficient. I came across this post: ...
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### Is it possible to ensure global convergence of a fixed point iteration?

Suppose I have a fixed point iteration of the form $$x_{n+1}=f(x_n).$$ Suppose further that after some initial testing, I found that it does not converge to an a priori known fixed point $x^*$. I ...
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### 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 ...
1answer
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### Conjugate Gradient for singular 2D poisson finite element with Neumann Boundary Conditions

Heavily edited question after I realised partly what the problem was I have programmed a simple 2D square finite element solution to the Poisson equation $-\Delta u = f$ The source function ...
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### Testing the time dependent Schrodinger Equation with an analytical solution?

I am numerically solving the Schrodinger Equation in 1D first and in higher dimension later, but I want to know the convergence rate of my numerical solver in different grid size and numerical methods....
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### $O(h^2)$ convergence for Elliptic PDE

I am trying to solve an elliptic PDE in 2-D: $$-\nabla^{2} u = 20tanh(10x-5)(10-10tanh(10x-5)^2) = f$$ I know that the solution is $u = tanh(10x-5)$ but I am unable to get $O(h^2)$ solution with a ...
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### How to test convergence of an algorithm for constrained optimization

I am applying an iterative method (projected newton) to an optimization problem. Theoretically, the method should converge linearly. I would greatly appreciate it if you could share how should I test ...
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### Computing rate and order of convergence

this is a follow up question to Convergence rate vs convergence order I guess the whole confusion about how rate vs. order of convergence also came from the implementations I saw. For example, for ...
2answers
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### How to choose the number of random points in Monte Carlo simulations?

I am struggling with convergence criteria when performing a Monte carlo simulation on a uniform distribution. Any help would be much appreciated ! Say I want to sample uniformly a 1D interval (for ...
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### Does Lanczos have trouble with large matrix elements?

I have a large, yet very sparse, matrix that I'd like to diagonalize. Both my own Lanczos implementation and the ARPACK that's built in with scipy fail to converge properly, though. I know that my ...
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### Normalizing error when data passes through zero

Given a time-dependent measure of error, $e(t)$, and a time-dependent function to normalize it with, $f(t)$, one would report the normalized error as $$E(t) = \left|\frac{e(t)}{f(t)}\right|.$$ ...
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### Does energy decrease with basis set size in density functional theory?

Based on the variational principle, one might expect that the ground state energy of a density functional theory (DFT) calculation will decrease as the basis set size increases. (As I understand it, ...
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### Best way to check if SOR solution has converged for 2d matrix

I have written a SOR algorithm to solve the Laplace equation on a 2d grid. The outside of the grid is fixed at 0 and the central square is fixed at 10. I can obtain the fully converged solution for ...