On Fri, Apr 11, 2008 at 8:00 PM, Alex Ghitza <[EMAIL PROTECTED]> wrote: > > -----BEGIN PGP SIGNED MESSAGE----- > Hash: SHA1 > > Hi, > > There are some inconsistencies in linear algebra over fields like CDF. > Some trouble was already reported in #2256. The following is another issue: > > sage: M = matrix(CDF, 2, 2, [(-1 - 2*I, 5 - 6*I), (-2 - 4*I, 10 - 12*I)]) > sage: M.is_invertible() > True > sage: M.determinant() > 5.3290705182e-15 + 1.7763568394e-15*I > sage: M.inverse() > [ 1.01330991616e+15 - 2.58956978574e+15*I -5.06654958079e+14 + > 1.29478489287e+15*I] > [ 5.62949953421e+14 + 5.62949953421e+14*I -2.81474976711e+14 - > 2.81474976711e+14*I] > > So because of roundoff errors, Sage thinks that we have an invertible > matrix. But the code for echelon_form knows that it's not invertible: > > sage: M.echelon_form() > [ 1.0 1.4 > + 3.2*I] > [-2.22044604925e-16 - 4.4408920985e-16*I > ~ 0] > sage: M.rank() > 1 > > This behavior is inconsistent. Either there is enough precision to > decide that the matrix is invertible, or there isn't. Doing this in two > different ways should not yield two mutually exclusive answers. > > Similar problems occur over CC. > > I hope it's clear by now that I have no idea how CDF and friends > actually work, so I don't know what would be a good solution or even > whether such a solution should exist. My initial guess would be that > finding the determinant requires more computations so there is more > precision loss than in computing the echelon form; in that case, maybe > over such fields we should not base is_invertible() and inverse() on > determinant() but rather on echelon_form().
This is now #2932: http://trac.sagemath.org/sage_trac/ticket/2932: Just looking at the value of M.rank() should be enough, I think. Also, M.rank() seems much faster than computing the determinant: {{{ sage: M = matrix(CDF, 2, 2, [(-1 - 2*I, 5 - 6*I), (-2 - 4*I, 10 - 12*I)]) sage: %timeit M.rank() 1000000 loops, best of 3: 676 ns per loop sage: %timeit M.det() 100000 loops, best of 3: 2.81 µs per loop }}} Timings are comparable over ZZ,QQ,RR. didier > > Note also that this problem does not seem to occur over RR or RDF. > > Best, > Alex > > > - -- > Alexandru Ghitza > Assistant Professor > Department of Mathematics > Colby College > Waterville, ME 04901 > http://bayes.colby.edu/~ghitza/ > -----BEGIN PGP SIGNATURE----- > Version: GnuPG v2.0.7 (GNU/Linux) > Comment: Using GnuPG with Mozilla - http://enigmail.mozdev.org > > iD8DBQFH//uydZTaNFFPILgRArYKAJ4r9QOJNYSSCgVzSutUW7gEfcv3uQCfd3aj > nYV2EFH6znwShoGL7q2BjRY= > =74HR > -----END PGP SIGNATURE----- > > > > --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
