It is well known that (square) Vandermonde matrices are badly conditioned (e.g. Gautschi: "How Unstable Are Vandermonde Systems").

But what about non square Vandermonde matrices?

This plot shows the average condition number vs. number of rows for a non-square Vandermonde matrix with 3 columns:

Condition Number vs. rows

It is interesting to see that the condition number is very high for a small number of rows but becomes small when the number of rows becomes large (much larger than columns).

For reference, this plot has been generated with the following MATLAB Code:

for trial=1:trials
    for N=1:1000
        V = fliplr(vander(randn(N,1)));
        V = V(:,2:4);
        conds(trial,N) = cond(V);
semilogx(1:1000, 10*log10(mean(conds)));

Why is this the case and can this be explained in an analytical way? Instead of condition number, I would also be happy with an argument based on the the determinant of $V^T V$

  • 1
    $\begingroup$ What's your definition of 'non-square Vandermonde matrix'? This older MO question has a definition, but I'd like to confirm whether it agrees with yours: mathoverflow.net/q/155384/55904 $\endgroup$ – Semiclassical Aug 24 '16 at 21:21
  • $\begingroup$ The definition is: Have nodes $x_i$, $i=1,\dots, N$ as first column. The $l$-th colum is $x_i^l$ and the last colum is $x_i^L$ where $N > L$ (i.e. more rows than columns) $\endgroup$ – divB Aug 24 '16 at 21:39
  • $\begingroup$ The definition is similar to the one you referenced. In my case it's just the transpose (I have the nodes as columns whereas the reference as rows) and I do not normalize the first row (i.e. in my case the first row is not the 0th power, i.e. 1 but the power of 1 - the nodes themselves) $\endgroup$ – divB Aug 24 '16 at 22:50

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