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.

learn more… | top users | synonyms

4
votes
1answer
98 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 ...
5
votes
1answer
71 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 ...
1
vote
1answer
45 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 ...
4
votes
1answer
154 views

convergence of unconstrained convex optimization

I encounter an optimization problem. The simplified version is like following: Denote function $F(x):\mathbf{R}^n\rightarrow\mathbf{R}$, where $F(x)$ is a smooth lower bounded convex function (i.e. ...
7
votes
0answers
42 views

DIIS method to accelerate SCF convergence for stretched geometries

I am implementing from scratch an Hartree-Fock calculation in the STO-3G basis set to perform Born-Oppenheimer molecular dynamics. I have a Restricted Hartree-Fock procedure that can reproduce very ...
0
votes
0answers
28 views

maltab ode solver- user defined criteria to stop calculations

is there a way to add a user defined convergence criteria to an ode solver so that the solution is stopped? I know that Matlab uses absolute and relative tolerances but would that suffice in solving ...
2
votes
2answers
96 views

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 ...
1
vote
2answers
101 views

Finding rate of convergence by curve fitting in Matlab

I have some data: number of nodes $N$ and error in energy norm corresponing to it. I have seen in some references that the rate of convergence is reported by $$\| u-u_h\| _E=CN^{\alpha} $$ How can ...
1
vote
0answers
25 views

Application of vector extrapolation methods to convergence to a steady state solution

I'm working on a fluid solver using dual-time stepping and everything works really well, except the convergence in pseudo-time is slow. I'd like to accelerate the convergence. I know multigrid methods ...
6
votes
0answers
56 views

Accelerated convergence for Sparse NMF

In the Non-Negative Matrix factorization (NMF), you basically compute an approximation of a given matrix $V \in \mathbb{R}_{+}^{n \times m}$ into matrices $W$ and $H$ such that $W \in ...
4
votes
0answers
57 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. ...
1
vote
1answer
69 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 ...
0
votes
0answers
59 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 ...
3
votes
1answer
110 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: ...
3
votes
2answers
208 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 ...
9
votes
2answers
184 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 ...
0
votes
1answer
242 views
1
vote
0answers
56 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) ...
6
votes
2answers
205 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 ...
4
votes
1answer
235 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; ...
1
vote
0answers
71 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 ...
1
vote
0answers
33 views

What is the Radius of Convergence for analytic functions? [closed]

The radius of convergence of any power series can be found by simply using the root test, ratio test etc. I am confused as to how to find the radius of convergence for an analytic $f$ such as ...
0
votes
1answer
139 views

Temperature dependent 1-d conduction in Python?

I'm trying to write a Python code that is a numerical solver for 1-d heat conduction (using FVM) with a temperature dependent thermal conductivity. The solver has three functions I need to iterate ...
0
votes
0answers
88 views

Convergence of a DASPK depending on DAE formulation

I have a system of non-linear DAE and I noticed that the system does not converge if some of the equations are not differentiated. For example, if the control volume equation is represented as this: ...
3
votes
2answers
698 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 ...
0
votes
1answer
57 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 ...
1
vote
0answers
69 views

Iterative algorithm prove precise conditions for convergence

Question: Consider the iterative improvement algorithm below. Starting with $Az_i = r_i$ and $(A + E)\hat{z}_i = r_i$ derive a formula showing how the absolute error in the $(i + 1)^{st}$ iterate ...
2
votes
3answers
261 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)} = ...
3
votes
2answers
91 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 ...
2
votes
1answer
1k views

The definition of asymptotic convergence?

What is the difference between convergence and asymptotic convergence? Why say the convergence is asymptotic?
3
votes
2answers
187 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?
3
votes
1answer
155 views

steady state solution from parabolic problem vs solution of elliptic problem

My question is related, but not a duplicate of Asymptotic convergence of the solution to a parabolic pde to the solution of an elliptic pde Suppose I solve the parabolic PDE: $u_t = \Delta u + ...
2
votes
2answers
336 views

What are some reasons that Conjugate Gradient iteration does not converge?

I would greatly appreciate it if you could share some reasons the Conjugate Gradient iteration for Ax = b does not converge? My matrix A is symmetric positive definite. Thank you so much! Edit with ...
2
votes
1answer
590 views

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 ...
7
votes
1answer
202 views

Finite element convergence rates for mixed problems

I've coded up a Stokes Flow problem using finite elements and am in the process of verifying that it works. I'm just not sure what convergence rate I should be expecting as I globally refine the mesh. ...
1
vote
0answers
72 views

Stationary phase approximation for an integral with infinity saddle points?

I need a hand with the numerical evaluation, in Mathematica, for this integral: $$f(t)=\int_{-\infty}^\infty Exp\{it(\omega_H-\omega_l-\omega_k) - \sum _{j\neq(l,k)} S_j [1-e^{-it\omega_j}]\}\, dt$$ ...
1
vote
0answers
237 views

BFGS Fails to Converge

The model I'm working on is a multinomial logit choice model. It's a very specific dataset so other existing MNLogit libraries don't fit with my data. So basically, it's a very complex function ...
3
votes
0answers
37 views

Solving ODE boundary problem with additional conditions

I need to solve two point bondary problem for ODE. The solution of the problem itself is very easy. The real issue is that when solving arising non linear system of equations it always converges to ...
1
vote
1answer
125 views

Problem with convergence of Jacobi iterative algorithm

I'm dealing with Jacobi iterative method for solving sparse system of linear equations. For small matrices it works well and gives right answers even if matrix is not strictly diagonal dominant, ...
4
votes
0answers
144 views

Conjugate residual/gradient convergence checking in practice

Let's say we want to solve $Ax=b$ ($A$ symmetric positive /semi/definite) with the conjugate residual/gradient method. $A$ comes from FEM where the mesh is being refined. The exact solution is $x_*$ ...
11
votes
0answers
1k 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 ...
4
votes
1answer
181 views

Multigrid stops converging when more grid levels are used

I'm having a problem with multigrid code I wrote. If I solve Laplace's equation in 2D and use more than 5 grid levels, the V-cycles stop converging after a few cycles (see below, convergence factor > ...
1
vote
1answer
2k views

Numerically determining convergence order of Euler's method

I need to numerically determine the convergence order of Euler's method for various step-sizes. I am unsure how to go about doing this. Here is the question: Problem statement: $\frac{dy}{dt}=\alpha ...
13
votes
3answers
247 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 ...
3
votes
1answer
132 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 ...
0
votes
0answers
71 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} = ...
2
votes
1answer
83 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 ...
4
votes
1answer
132 views

What is the origin of the preasymptotic convergence behaviour in FEM?

When you have fine-scale features (e.g. boundary layers) in the solution, its FEM approximation on coarse meshes converge at strange apparent rates. Looking at Cea's lemma, is this behaviour because ...
1
vote
3answers
119 views

Guess the final term of a converging series [closed]

I have a non-linear equation that converges, and reaches suitable accuracy after around 20 steps, however each step is very expensive to calculate. The series are never quite the same, but they are ...
3
votes
0answers
74 views

analytic or numeric integral of diverging function

I'm trying to carry out the following integral numerically $$\int_{r_\mathrm{in}}^{r_\mathrm{out}} \Sigma\left(r'\right) \frac{r'}{r} \left( \frac{1}{r-r'}\, E(L) + \frac{1}{r+r'}\, K(L) \right) ...