Krdytkyu

I read from the an excellent recent paper on solving ill-conditioned Vandermonde matrix, they are outperforming the state-of-the-art, i.e., Bjorck and V. Pereyra method

An interesting quote from the paper is that Matlab cond is not trustworthy when the output condition number exceeds machine precision.

Example 3.2. The best known example of ill–conditioned Cauchy matrix
is the Hilbert matrix, Hij = 1/(i + j − 1). For instance, the
condition number of the 100 × 100 Hilbert matrix satisfies κ2(H) >
10150. Ill-conditioning is not always obvious in the sizes of its entries – the entries cond(hilb(100)) returns
ans=4.622567959141155e+19. One should keep in mind that the matrix of
the 100 × 100 Hilbert matrix range from 1/199 ≈ 5.025 · 10−3 to 1.
Moreover, in Matlab, condition number is a matrix function with its
own condition number. By a result of Higham [21], condition number of
the condition number is the condition number itself, meaning that our
computed condition number, if it is above 1/ε (in Matlab,
1/eps=4.503599627370496e+15), it might be entirely wrongly computed.
This may lead to an underestimate of extra precision needed to handle
the ill–conditioning.

My question:

Is it in effect, equivalent to say: any computed results from a double-precision machine above 1e16 is not truthworthy?, which seems natural.

What worries me

Note that in equation (4.1) of that paper, they computed the condition number of ill-conditioned matrix (two of those are 10^76 and 10^21, both much larger than 1e16) and continue to use those numbers for further arguments.

We choose the scaling in the l2 norm. The
relevant condition numbers, estimated using the Matlab’s function
cond() are as follows cond(Vm) ≈ 8.9 · 10^76, cond(V(r) m ) ≈ 3.1 ·
10^7, cond(V(c) m ) ≈ 3.0 · 10^21

What I have done

I did some test, and I don't know how to compute condition number of extremely ill conditioned matrix that can give me condition numbers above 1e20.

Further question:

Can we increase Matlab double precision to much much higher precision? So cond is trustworthy? I tried vpa as a wrap up for cond(hilb(100)) and it is not working as I want.