# Tag Info

Accepted

Accepted

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

Let's examine the one-dimensional three-point stencil case in detail, because I think it's important to be clear just how this behaviour arises, and what it means to set a point to a certain value in ...
Accepted

### Lack of quadratic convergence in Newton's method

Yes: See Higham's book "Accuracy And Stability of Numerical Algorithms", second edition, chapter 25: Nonlinear Systems and Newton's Method. In particular, see the section on the "limiting residual" ...

### Strong coupling of a non-linear multiphysic problem: failure with Newton Raphson method

The issues you're running into now are not a failing of Newton-Raphson, but a question of coupling. You're doing iterated sequential coupling -- solving each equation sequentially and then iterating ...
Accepted

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

If your matrix is symmetric, positive definite, the CG method may converge slowly, but it converges for $n\to\infty$. The only reason it does not converge on a computer are round-off errors, in ...
Accepted

### Global convergence in trust region algorithm

Standard terminology in nonlinear (i.e., derivative-based) optimization is that "global convergence" means "convergence to a stationary point no matter where you start from" (where the limit may in ...

### Which absolute and/or relative stopping criteria do use for Newton's method?

Some of the confusion comes about because mathematicians often consider artificial functions that happen to have size $O(1)$ whereas practitioners use functions and variables that just happen to be ...

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

I don't think there's enough information to give a heuristic like, "in general, it takes $k$ iterations for differential evolution to reach a global optimum". First, we don't know how many iterations ...
Accepted

### Finding rate of convergence by curve fitting in Matlab

First of all, rates of convergence are usually given in the form $$\|u-u_h\| \leq C N^\alpha,$$ rather than equality. Furthermore, rates are asymptotic, i.e., only have to hold for $N\to \infty$. ...

### Scaling/nondimensionalization for numerical optimization

One thing that makes your non-dimensionalized ODE confusing is that you use the same symbols for dimensionalized and nondimensionalized variables, even though they are different variables. Consider ...

### Analytical convergent sequence and numerical divergent sequence

In my opinion, an example could be the calculation of the minimal solution of a three term recurrence relation (TTRR) $$y_{n+1} +a_{n}y_{n}+b_{n}y_{n-1} = 0, \quad n=1,2,3,\ldots$$. For example, the ...

### Accuracy issues with Arpack in Julia for eigenvalues of smallest magnitude

Looks like everything is working relatively OK? Your matrix is of order 1e10, so residuals of 1e-4 are actually close to machine precision. The convergence criterion is indeed violated, but not by ...

### Why in scientific papers convergence of finite difference and finite volume schemes is tested using multiple norms ($l_1$, $l_2$ and $l_{\infty}$)?

Both your questions have been addressed in the answer of @WolfgangBangerth. Here is an applied example of the behaviour of the different norms in the context of polynomial approximation (which is for ...

### CFL condition for variable coefficients

The CFL number (or simply Courant number) is defined locally: $\nu = u(x,t) \frac{\Delta t}{\Delta x}$. The velocity may vary in time and space and the grid may be non-uniform (in $x$ and/or $t$). ...