Thanks for the replay Wolfgang. And no disrespect to Terry Tao. I had never heard of him before these articles. I guess I am pretty disconnected from the math community!
You successfully answered all my questions; however, you now have me curious why the physics community would want to know all (or a larger portion of) the eigenvalues. If they're using these methods to somehow characterize the wide frequency range of light or radiation, I can imagine that would get nasty very quickly! Kind regards, Earl On Sun, Nov 17, 2019 at 10:43 AM Wolfgang Bangerth <[email protected]> wrote: > > Earll, > > > I apologize this is not directly Deal.ii related, but I wanted to share > it in > > case it contains something enlightening for deal.ii developers. > > > > A friend of mine in the physics community shared this article: > > [...] > > > > My knowledge of eigenvector and eigenvalue calculation is very rusty, > but I > > understand they are claiming to have simplified eigenvector calculation > under > > certain conditions. Does anyone know what this means in a practical > sense? Is > > this clickbait? Does anyone understand this article well enough to say > if it > > has computational implications outside quantum mechanics? > > I'm no eigenvalue expert either, but it's an interesting formula. In > general, > if Terence Tao speaks, it's worth listening :-) > > As for practical impact, I think this comment is useful: > > https://terrytao.wordpress.com/2019/08/13/eigenvectors-from-eigenvalues/#comment-529043 > > But at least as far as the community on this mailing list is concerned, > it's > worth keeping in mind two issues: > > * The matrices we have are sparse; that makes a lot of things more > complicated, but also some much easier. > > * The matrices for which we consider eigenvalues are generally > finite-dimensional approximations of infinite-dimensional operators. That > means that in general only the matrix eigenvalues at either the low or the > high end of the spectrum are good approximations of the operator > eigenvalues, > and those are the ones we're generally interested in. In other words, we > almost never compute all eigenvalues of a matrix because most of them are > meaningless. So we only care about a few eigenvalues and corresponding > eigenvectors (and have efficient Krylov space methods to compute them) -- > but > then a formula that requires us to know *all* eigenvalues of a matrix > (plus > all eigenvalues of all of the M matrices) is not going to be of great help > to > our community. > > Best > W. > > -- > ------------------------------------------------------------------------ > Wolfgang Bangerth email: [email protected] > www: http://www.math.colostate.edu/~bangerth/ > > -- The deal.II project is located at http://www.dealii.org/ For mailing list/forum options, see https://groups.google.com/d/forum/dealii?hl=en --- You received this message because you are subscribed to the Google Groups "deal.II User Group" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/dealii/CAK30MZ6VXR6zenSArvpqQfRZFEdWQiJR_RBbaAioTscSzADk%3DA%40mail.gmail.com.
