John, certainly, over exact field you don't want to create unnecessary square roots. (actually, I would argue against normalisation in fields like QQbar, as division is expensive there...) Dima
On Apr 20, 7:42 pm, John Cremona <john.crem...@gmail.com> wrote: > I would say: over an inexact field like R or C then it is sensible to > normalize as Dima suggests (norm 1) rather than making any one > nonzero coordinate 1. But over exact fields (e.g. finite fields, > number fields) it does make perfect sense to normalise to the first > (or last?) nonzero coordinate is 1. And over a field such as Q which > is the fraction field of a uniqu factorization domain (e.g. Z) one > coule argue for normalising so that the coordinates were coprime > integers. > > In summary: what is a sensible or desirable normalisation depends a > lot on what the field is and what sort of mathematics you are doing! > > John > > On 20 April 2010 12:37, Dima Pasechnik <dimp...@gmail.com> wrote: > > > > > Dan, > > > indeed, it's not too bad to normalize to norm 1, say, but it is quite > > bad to normalize a given coordinate to 1. > > I cc this to sage-devel > > > Best, > > Dima > > > On Apr 18, 11:21 am, Dan Drake <dr...@kaist.edu> wrote: > >> On Sat, 17 Apr 2010 at 07:50PM -0700, Dima Pasechnik wrote: > >> > On Apr 18, 3:29 am, William Stein <wst...@gmail.com> wrote: > >> > > The test should be rewritten in a way to allow for either sign, since > >> > > either is correct... or we should re-normalize all the numerical > >> > > eigenvectors so the first nonzero entry is 1. I like the idea to > >> > > normalize. > > >> > no, this is a recipe for disaster. If the 1st nonzero entry is close > >> > to 0, the division by it will blow the big entries off. Renormalize > >> > to have the 1st non-0 entry positive, OK. But no division, please, > >> > unless really necessary. > > >> I'm not necessarily arguing for normalization, but I recall that Matlab > >> normalizes eigenvectors to have length 1. It seems like that kind of > >> normalization won't behave badly when an entry is very tiny, since the > >> difference in length is correspondingly tiny. > > >> Also, Matlab is very widely used for numerical mathematics, and has been > >> for a long time. If that kind of normalization was unstable, it's > >> probably unlikely that it would continue to be used. (Although IANANAE > >> -- I am not a numerical analysis expert.) > > >> (I find it amusing that IANANAE contains "NAN"...) > > >> Also, this discussion probably belongs on sage-devel, as it directly > >> concerns, well, development. :) > > >> Dan > > >> -- > >> --- Dan Drake > >> ----- http://mathsci.kaist.ac.kr/~drake > >> ------- > > >> signature.asc > >> < 1KViewDownload > > > -- > > To post to this group, send an email to sage-devel@googlegroups.com > > To unsubscribe from this group, send an email to > > sage-devel+unsubscr...@googlegroups.com > > For more options, visit this group > > athttp://groups.google.com/group/sage-devel > > URL:http://www.sagemath.org > > -- > To post to this group, send an email to sage-devel@googlegroups.com > To unsubscribe from this group, send an email to > sage-devel+unsubscr...@googlegroups.com > For more options, visit this group athttp://groups.google.com/group/sage-devel > URL:http://www.sagemath.org -- To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
