2
$\begingroup$

I'm studying for an exam and I came into the following question regarding the use of eigenvalues to estimate a solution. I'm having trubles understating what the request is, can somebody give me a hint? I can't find a good relationship, without computing the eigenvectors, between the solution of a problem and the eigenvalues of the matrix.

underlying of the confusing request

$\endgroup$
6
  • 1
    $\begingroup$ Hint 1: suppose A is a 1x1 matrix, how does eigenvalue relate to entr(ies) of A? Can you estimate the error from eigenvalue? Hint 2: now suppose A is a diagonal matrix, can you write solution in terms of eigenvalues and estimate the error? $\endgroup$ Commented Oct 12 at 19:38
  • 1
    $\begingroup$ I think the answer is less clear. It could be as Yaraslov says and you are meant to use an FFT to explicitly compute the solution to compare. It could also be that you are meant to use the spectrum of $A$ to estimate the error $\|u^*-u\|$ from the residual $\|f-Au\|$, or that you are meant to use the spectrum and properties of preconditioned conjugate gradient to get a finer estimate. I would ask your professor $\endgroup$ Commented Oct 12 at 20:01
  • 1
    $\begingroup$ Here's what chatgpt says which looks reasonable chatgpt.com/s/t_68ec094264388191b1f21a6339cec874 $\endgroup$ Commented Oct 12 at 20:02
  • $\begingroup$ Thanks to both of you $\endgroup$ Commented Oct 13 at 19:34
  • $\begingroup$ The estimate from ChatGPT (i.e., just computing the condition number) ignores the use of the Krylov method. Also, could you specify what $f$ is in this problem? That drastically changes the convergence properties. I would again suggest asking your professor $\endgroup$ Commented Oct 13 at 20:46

0

Reset to default

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.