Jason Grout wrote: > > I don't think it's an issue of irrational versus rational. It's > numerical precision and inexact floating point numbers. This matrix is > terribly ill-conditioned. It is right on the border line between being > invertible or not, numerically speaking:
No, it isn't. *My* matrix contains nine integer multiples of 0.1, which any high school student can show represents a linearly-dependent system in a few seconds. > sage: m.change_ring(RDF).SVD()[1] > > [ 0.772642968023 0.0 0.0] > [ 0.0 0.450580563234 0.0] > [ 0.0 0.0 3.13289758759e-17] > > As you probably know, the ratio between the smallest and largest > eigenvalues being so high gives us an indication that this matrix is > really a messy one numerically, and deserves the strictest of attention. I don't know that. Why would I know that? Why would I compute the smallest and largest eigenvalues when I just want the RREF of the thing? The original matrix is *not* messy. It's only messy once SAGE handles it in a messy way. What's worse is that there is no indication that the results of echelon_form() are incorrect. If I didn't know that matrix was singular, I would have happily continued my work with results that were essentially made up. Sure, I could check each matrix by hand, but /that's why I'm using SAGE in the first place/. > If we are allowed to do the computations with exact arithmetic (e.g., > using fractions instead of decimals), then we can compute its rank and > echelon form exactly. You see the same problems in other math software > too, with different matrices. If pressed, I could come up with an a > very nice-looking matrix example from a linear algebra textbook problem > that matlab totally gave the wrong answer to because it was a very > ill-conditioned matrix. I concede that MATLAB et al. can under certain circumstances produce incorrect results, but that doesn't mean that SAGE should give up on correctness completely. No other software fails on this example. --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to sage-support-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-support URL: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
