19
votes
Central differencing scheme for second derivative leads to ill-conditioning
TL;DR: The continuous operator exhibits this behavior, any faithful discretization will just inherit it.
Deeper cut: If you look at the spectrum (eigenpairs) of the continuous operator $\frac{d^2}{dx^...
- 4,862
10
votes
Accepted
Analytical convergent sequence and numerical divergent sequence
Jean-Michel Muller, et. al., "Handbook of Floating-Point Arithmetic 2nd ed.", Birkhäuser 2018, gives the following example due to Muller, specifically constructed to deliver incorrect results with ...
- 1,865
7
votes
What is a well-posed problem?
Continuous dependence on initial conditions means that if for every $\epsilon > 0$ there exists a $\delta(\epsilon) > 0 $ such that
$$\Vert x_0 - \widetilde{x_0} \Vert \leq \delta(\epsilon) \...
- 1,083
6
votes
Accepted
Why are the round-off errors when solving the linear system $Ax = b$ of order $\varepsilon_\text{mach} x_j$?
The previous and next IEEE machine numbers to $\alpha_j$ are at a distance $\approx |\alpha_j| \varepsilon_{mach}$ from each other; hence $fl(\alpha_j)$ (the closest machine number to $\alpha$) is at ...
- 11.3k
6
votes
Analytical convergent sequence and numerical divergent sequence
In my opinion, an example could be the calculation of the minimal solution of a three term recurrence relation (TTRR) $$y_{n+1} +a_{n}y_{n}+b_{n}y_{n-1} = 0, \quad n=1,2,3,\ldots$$.
For example, the ...
- 6,179
3
votes
What is a well-posed problem?
Some of the classic examples of ill-posed problems are to infer the coefficients of a PDE from measurements of the solution.
For a specific example, consider the Poisson problem
$$-\nabla^2 u = f$$
on ...
- 10.3k
2
votes
Condition number of two perburbation matrix regarding limit and quadtrature integration rules
1. For a matrix $A$ with distinct eigenvalues, adding a perturbation $\delta A$ results [1] in a change to eigenvalues of magnitude (to first order)
$$ \delta\lambda_i = (X^{-1}\delta A X)_{ii}, $$
so ...
- 11.4k
2
votes
Accepted
Condition number of matrix and effects of round off errors
For sparse matrices $A$, that are a discretized version of operators in PDEs in FE, FV, or FD, you do know your sparsity pattern before you compute the actual entries. So, you usually
compute matrix ...
- 8,672
2
votes
Accepted
ill-conditioning
It might help to start with a concrete example of a function, like $f(x,y)=10^9 x^2+y^2$ (the example they use on p.26). I'm trying to guess what went wrong with your analysis, because you haven't ...
- 11.4k
1
vote
What is a well-posed problem?
I suspect you are confusing well-posed optimization problem (which you would want one solution to, regardless of initial guess) and well-posed differential equation (which you would want to be ...
- 1,848
1
vote
What is a relative condition number of a sum of positive values?
I don't think your own answer is correct, though my method somewhat differs in so far as I use a different norm than you - the Euclidean norm. My result:
The relative condition for the sum of two ...
- 111
1
vote
What is a relative condition number of a sum of positive values?
The mistake I made is that the Jacobian in the infinity norm is equal to 1, not $n$. As such, the relative condition number of the sum is:
$$\kappa=1$$
and as such is always well-conditioned, no ...
- 2,930
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