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# 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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1answer
111 views

### What should I put on the paper to show the correctness and convergence of my solution?

I am using FEM to do an assignment on a heat conduction problem on a complex domain, which needs me to get the variation of the temparature distribution subject to the variation of boundary conditions,...
1answer
195 views

### Problem with Richardson extrapolation method for weak convergence in SDE

I have implemented the Richardson extrapolation of the Euler-Maruyama method to 4th order, to estimate the moments of SDE. The Euler-Maruyama works, and I would expect the Richardson extrapolation to ...
0answers
128 views

### Are there any benefits of computable analysis to numerical algorithms

Computers can work only with computable numbers, while most of the algorithms are based on analysis of real numbers (real analysis). When I heard of the existence of computable analysis I ...
1answer
2k 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
434 views

### Are self-convergence tests reliable?

I'm developing a solver for solving linear hyperbolic equations of first order with respect to time and spatial derivatives. The formal order of accuracy of the solver must be 5 because I use 5th-...
1answer
71 views

### Expected number of steps before a global optimum is found with Simulated Annealing

I'm reading a technical report on Simulated Annealing: On the Convergence Time of Simulated Annealing, by Sanguthevar Rajasekaran. You may find it following this link. Given $G=(V, E)$ is the graph ...
1answer
206 views

0answers
191 views

### What causes periodic humps in residual plots?

When using many iterative methods, whether for solving linear systems, looking for steady-state convergence in CFD, etc., the semilog plot of the residual often shows "humps" as the residual decays. ...
1answer
221 views

### Convergence problem in iterative method

I am trying to solve two non-linear equations self-consistently in a Gummel loop. Sometimes (every once in a while), I get to a situation when the loop repeats itself with wrong solutions and a ...
0answers
199 views

### Oscillating convergence in my Resilient BackPropagation (RPROP) implementation

I have implemented in matlab a neural network that uses rprop's algorithm to update its weights. Strangely the error on the training set does not converge to a local minimum, but oscillates. Here is ...
1answer
774 views

### 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: ...
2answers
8k views

### Error in result of finite-difference approximation when refining

I have calculated the first derivative of following equation using Euler method (first order), Three point Finite Difference method (second order) and Four point Finite Difference method (third order)....
2answers
345 views

### How does weak convergence feel, numerically?

Consider, you have a problem in an infinite dimensional Hilbert or Banach space (think of a PDE or an optimization problem in such a space) and you have an algorithm that converges weakly to a ...
1answer
2k views

### scipy: what does scipy.integrate.odeint's Repeated convergence failures (perhaps bad Jacobian or tolerances)." error mean?

What does the following scipy.integrate.odeint error mean numerically? ...
1answer
232 views

### CFD: Doubt with time convergence in advection fully implicit upwind scheme

I'm trying to solve an advection - convection problem using an implicit upwind scheme - you can see here the finite difference discretization used. I start the model (built from scratch on Scilab) ...
3answers
401 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
298 views

### When and why is r./sum(r) not a good way to renormalize a vector in PageRank computation?

I experimented with the PageRank algorithm. When the number of pages is large, I encountered a situation when one formula for re-normalizing a vector (so that sum of its components is equal to 1; ...
0answers
226 views

### Am I using the incorrect implementation of the fast Chebyshev transform?

I was told that the fast Chebyshev transform has superior spectral convergence, but I am unable to verify its rumored convergence. I was given plots of its spectral convergence, where the signal's ...
0answers
46 views

2answers
4k views

### The rate of convergence for finite difference methods for Poisson's equation with piecewise constant data

I am solving the following PDE; $$\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \rho,$$ where $\rho(0.5,0.5) = 2$ (zero elsewhere), $0\leq x,y\leq1$ and the ...
1answer
116 views

### Actually calculating the rate of convergence of iteratvie methods when exact solution is unknown

When solving a system of nonlinear equations using iterative methods, the rate of convergence usually is defined by the following formula: (1) where x* is the exact solution. However usually we ...
0answers
184 views

2answers
128 views

### How does constraint resolution affect the stability/accuracy of numerical integration?

I understand some basic analysis techniques (local truncation error, global error, zero-stable, absolute stable, etc.) of numerical integration. But I find it hard to apply these techniques in ...
1answer
11k views

### The definition of asymptotic convergence?

What is the difference between convergence and asymptotic convergence? Why say the convergence is asymptotic?
2answers
629 views

### How many generations does it typically take for a differential evolution method to reach a global optimum?

For differential evolution methods in optimization, how many generations does it typically take to reach a global optimum? How do we know if the values are never going to converge?
1answer
305 views

3answers
304 views

### Computing slightly oscillatory series to high precision?

Suppose I have the following interesting function: $$f(x) = \sum_{k\geq1} \frac{\cos k x}{k^2(2-\cos kx)}.$$ It has some unpleasant properties, like its derivative not being continous at rational ...
1answer
149 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
87 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} = 0\...
1answer
156 views

### 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 ...