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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^...
rchilton1980's user avatar
  • 4,906
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 ...
njuffa's user avatar
  • 1,875
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) \...
Dan Doe's user avatar
  • 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 ...
Federico Poloni's user avatar
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 ...
GertVdE's user avatar
  • 6,149
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 ...
Daniel Shapero's user avatar
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 ...
Kirill's user avatar
  • 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 ...
Anton Menshov's user avatar
  • 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 ...
Kirill's user avatar
  • 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 ...
Johan Löfberg's user avatar
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 ...
Stijn D'hondt's user avatar
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 ...
Ondřej Čertík's user avatar

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