@tsunamiBTP Look up "catasthrophic cancellation" and especially the Wikipedia article "Loss of significance." The case at hand is an example of catasthrophic cancellation: the substraction of two quantities whose difference is many orders of magnitude smaller than the quantities.

@Math Pi Euler Try copy-and-pasting Tom's code. I think that you may have been entering it manually.

@Math Pi Euler Yeah, thanks, I figured that out by now. Anyway, I think that the Answer by Tom Leslie is sufficient for your purpose. If not, let me know.

What are the limits of summation?

Try typing your expressions here the same way as you would in Maple.

You say "verify the following formula" but I don't see any formula. You'll need to provide more information.

Update: The above comment applied to an earlier version of the Question.

@macmp91 So, we see that the minimal value of what you call w^2 eventually becomes negative. Why not use that as the criterion to stop your while loop? Don't take the sqrt in procedure Fn; take it afterwards.

@macmp91 Let's forget about the singular values for now. I don't know if they're appropriate for your problem. Why do you take the square root of the eigenvalues? Why not just stop your while loop when the minimal eigenvalue is negative?

@dbain1 The numerical integral is very difficult to evaluate for some values of p, such as 0.15. Your integrand looks a bit strange. Did you perhaps intend for it to be the absolute value of the expression that is to be raised to the 1/p power? If I include abs thus, there is no problem doing the numeric integration. Without abs, we end up raising negative values to fractional powers, obtaining imaginary results.

This is a topic that has been much argued on MaplePrimes. The appropriate question is How much effort (or time or expense) should be expended determing whether a given expression is identically zero when that problem is known to be in general undecidable? You may adjust that level of effort using TestZero. But no matter what you set it to, there will always be some singular matrix whose singularity can't be detected. Thus, if you call it a bug, there will necessarily always be a bug. I'd say that somthing that is mathematically impossible to fix shouldn't be called a bug. See the Wikipedia article "Richardson's Theorem" https://en.wikipedia.org/wiki/Richardson%27s_theorem

If you could provide a reference for an algorithm, I could probably implement it in Maple. I'm browsing the Wikipedia article "Forward Error Correction" https://en.wikipedia.org/wiki/Forward_error_correction, but the topic is too large.

Without going into much detail, I see that your final plot command fails because there is no numeric value for q. Is your q supposed to have a value?

@mapleatha Sorry, I misread your Question. I thought that you were getting unevaluated output and the blue was what you were expecting. My fault, totally. This Answer can just be ignored.

Is the domain of the function a finite set?

@kfli For some reason that I can't figure out, the freeze command is intentionally "crippled" to not work on names such as x[n]. This crippling is obvious in the very brief code that you can see via showstat(freeze). But it will work on x(n), which is considered a  function rather than a name. So try changing all x[n] to x(n). If you want these to display as subscripted, that can still be done. You can freeze a whole batch of expressions in a single command in a variety of ways. One example is

assign(seq(x(k)= freeze(y(k)), k= 1..9));

Let me know how that works for you.

@ecterrab I think that your impression that the OP is trying to differentiate wrt a function may be based on my example rather than on the OP's code. I just used that example because I didn't have the patience to decipher the OP's convoluted alias commands.